├── MIT18_650F16_PSet1.pdf ├── MIT18_650F16_PSet10.pdf ├── MIT18_650F16_PSet11.pdf ├── MIT18_650F16_PSet2.pdf ├── MIT18_650F16_PSet3.pdf ├── MIT18_650F16_PSet4.pdf ├── MIT18_650F16_PSet5.pdf ├── MIT18_650F16_PSet6.pdf ├── MIT18_650F16_PSet7.pdf ├── MIT18_650F16_PSet8.pdf ├── MIT18_650F16_PSet9.pdf ├── PS1 ├── ProblemSet1.aux ├── ProblemSet1.bbl ├── ProblemSet1.blg ├── ProblemSet1.log ├── ProblemSet1.pdf ├── ProblemSet1.synctex.gz └── ProblemSet1.tex ├── PS10 ├── PropblemSet10.aux ├── PropblemSet10.log ├── PropblemSet10.out ├── PropblemSet10.pdf ├── PropblemSet10.synctex.gz └── PropblemSet10.tex ├── PS11 ├── ProblemSet11.aux ├── ProblemSet11.log ├── ProblemSet11.out ├── ProblemSet11.pdf ├── ProblemSet11.synctex.gz └── ProblemSet11.tex ├── PS2 ├── Problem_set2.aux ├── Problem_set2.log ├── Problem_set2.out ├── Problem_set2.pdf ├── Problem_set2.synctex.gz └── Problem_set2.tex ├── PS3 ├── ProblemSet3.aux ├── ProblemSet3.log ├── ProblemSet3.pdf ├── ProblemSet3.synctex.gz └── ProblemSet3.tex ├── PS4 ├── ProbemSet4.aux ├── ProbemSet4.log ├── ProbemSet4.out ├── ProbemSet4.pdf ├── ProbemSet4.synctex.gz └── ProbemSet4.tex ├── PS5 ├── ProblemSet5.aux ├── ProblemSet5.log ├── ProblemSet5.pdf ├── ProblemSet5.synctex.gz └── ProblemSet5.tex ├── PS6 ├── ProblemSet6.aux ├── ProblemSet6.log ├── ProblemSet6.pdf ├── ProblemSet6.synctex.gz ├── ProblemSet6.tex └── img.jpg ├── PS7 ├── ProlemSet7_solustion.aux ├── ProlemSet7_solustion.log ├── ProlemSet7_solustion.pdf ├── ProlemSet7_solustion.synctex.gz └── ProlemSet7_solustion.tex └── README.md /MIT18_650F16_PSet1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet1.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet10.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet10.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet11.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet11.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet2.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet2.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet3.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet3.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet4.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet4.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet5.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet5.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet6.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet6.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet7.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet7.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet8.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet8.pdf -------------------------------------------------------------------------------- /MIT18_650F16_PSet9.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/MIT18_650F16_PSet9.pdf -------------------------------------------------------------------------------- /PS1/ProblemSet1.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | -------------------------------------------------------------------------------- /PS1/ProblemSet1.bbl: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS1/ProblemSet1.bbl -------------------------------------------------------------------------------- /PS1/ProblemSet1.blg: -------------------------------------------------------------------------------- 1 | This is BibTeX, Version 0.99dThe top-level auxiliary file: ProblemSet1.aux 2 | I found no \citation commands---while reading file ProblemSet1.aux 3 | I found no \bibdata command---while reading file ProblemSet1.aux 4 | I found no \bibstyle command---while reading file ProblemSet1.aux 5 | (There were 3 error messages) 6 | -------------------------------------------------------------------------------- /PS1/ProblemSet1.log: -------------------------------------------------------------------------------- 1 | This is pdfTeX, Version 3.14159265-2.6-1.40.20 (MiKTeX 2.9.7140 64-bit) (preloaded format=pdflatex 2019.9.10) 19 SEP 2019 17:00 2 | entering extended mode 3 | **./ProblemSet1.tex 4 | (ProblemSet1.tex 5 | LaTeX2e <2018-12-01> 6 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/base\article.cls" 7 | Document Class: article 2018/09/03 v1.4i Standard LaTeX document class 8 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/base\size10.clo" 9 | File: size10.clo 2018/09/03 v1.4i Standard LaTeX file (size option) 10 | ) 11 | \c@part=\count80 12 | \c@section=\count81 13 | \c@subsection=\count82 14 | \c@subsubsection=\count83 15 | \c@paragraph=\count84 16 | \c@subparagraph=\count85 17 | \c@figure=\count86 18 | \c@table=\count87 19 | \abovecaptionskip=\skip41 20 | \belowcaptionskip=\skip42 21 | \bibindent=\dimen102 22 | ) 23 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/geometry\geometry.s 24 | ty" 25 | Package: geometry 2018/04/16 v5.8 Page Geometry 26 | 27 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics\keyval.sty 28 | " 29 | Package: keyval 2014/10/28 v1.15 key=value parser (DPC) 30 | \KV@toks@=\toks14 31 | ) 32 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\ifpdf.st 33 | y" 34 | Package: ifpdf 2018/09/07 v3.3 Provides the ifpdf switch 35 | ) 36 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\ifvtex.s 37 | ty" 38 | Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) 39 | Package ifvtex Info: VTeX not detected. 40 | ) 41 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/ifxetex\ifxetex.s 42 | ty" 43 | Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional 44 | ) 45 | \Gm@cnth=\count88 46 | \Gm@cntv=\count89 47 | \c@Gm@tempcnt=\count90 48 | \Gm@bindingoffset=\dimen103 49 | \Gm@wd@mp=\dimen104 50 | \Gm@odd@mp=\dimen105 51 | \Gm@even@mp=\dimen106 52 | \Gm@layoutwidth=\dimen107 53 | \Gm@layoutheight=\dimen108 54 | \Gm@layouthoffset=\dimen109 55 | \Gm@layoutvoffset=\dimen110 56 | \Gm@dimlist=\toks15 57 | 58 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/geometry\geometry.c 59 | fg")) 60 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amsmath.sty 61 | " 62 | Package: amsmath 2018/12/01 v2.17b AMS math features 63 | \@mathmargin=\skip43 64 | 65 | For additional information on amsmath, use the `?' option. 66 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amstext.sty 67 | " 68 | Package: amstext 2000/06/29 v2.01 AMS text 69 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amsgen.sty 70 | " 71 | File: amsgen.sty 1999/11/30 v2.0 generic functions 72 | \@emptytoks=\toks16 73 | \ex@=\dimen111 74 | )) 75 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amsbsy.sty" 76 | Package: amsbsy 1999/11/29 v1.2d Bold Symbols 77 | \pmbraise@=\dimen112 78 | ) ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amsopn.st 79 | y" 80 | Package: amsopn 2016/03/08 v2.02 operator names 81 | ) 82 | \inf@bad=\count91 83 | LaTeX Info: Redefining \frac on input line 223. 84 | \uproot@=\count92 85 | \leftroot@=\count93 86 | LaTeX Info: Redefining \overline on input line 385. 87 | \classnum@=\count94 88 | \DOTSCASE@=\count95 89 | LaTeX Info: Redefining \ldots on input line 482. 90 | LaTeX Info: Redefining \dots on input line 485. 91 | LaTeX Info: Redefining \cdots on input line 606. 92 | \Mathstrutbox@=\box27 93 | \strutbox@=\box28 94 | \big@size=\dimen113 95 | LaTeX Font Info: Redeclaring font encoding OML on input line 729. 96 | LaTeX Font Info: Redeclaring font encoding OMS on input line 730. 97 | \macc@depth=\count96 98 | \c@MaxMatrixCols=\count97 99 | \dotsspace@=\muskip10 100 | \c@parentequation=\count98 101 | \dspbrk@lvl=\count99 102 | \tag@help=\toks17 103 | \row@=\count100 104 | \column@=\count101 105 | \maxfields@=\count102 106 | \andhelp@=\toks18 107 | \eqnshift@=\dimen114 108 | \alignsep@=\dimen115 109 | \tagshift@=\dimen116 110 | \tagwidth@=\dimen117 111 | \totwidth@=\dimen118 112 | \lineht@=\dimen119 113 | \@envbody=\toks19 114 | \multlinegap=\skip44 115 | \multlinetaggap=\skip45 116 | \mathdisplay@stack=\toks20 117 | LaTeX Info: Redefining \[ on input line 2844. 118 | LaTeX Info: Redefining \] on input line 2845. 119 | ) 120 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amscls\amsthm.sty" 121 | Package: amsthm 2017/10/31 v2.20.4 122 | \thm@style=\toks21 123 | \thm@bodyfont=\toks22 124 | \thm@headfont=\toks23 125 | \thm@notefont=\toks24 126 | \thm@headpunct=\toks25 127 | \thm@preskip=\skip46 128 | \thm@postskip=\skip47 129 | \thm@headsep=\skip48 130 | \dth@everypar=\toks26 131 | ) 132 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsfonts\amssymb.st 133 | y" 134 | Package: amssymb 2013/01/14 v3.01 AMS font symbols 135 | 136 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsfonts\amsfonts.s 137 | ty" 138 | Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support 139 | \symAMSa=\mathgroup4 140 | \symAMSb=\mathgroup5 141 | LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' 142 | (Font) U/euf/m/n --> U/euf/b/n on input line 106. 143 | )) 144 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics\graphicx.s 145 | ty" 146 | Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) 147 | 148 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics\graphics.s 149 | ty" 150 | Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR) 151 | 152 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics\trig.sty" 153 | Package: trig 2016/01/03 v1.10 sin cos tan (DPC) 154 | ) 155 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics-cfg\graphi 156 | cs.cfg" 157 | File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration 158 | ) 159 | Package graphics Info: Driver file: pdftex.def on input line 99. 160 | 161 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics-def\pdftex 162 | .def" 163 | File: pdftex.def 2018/01/08 v1.0l Graphics/color driver for pdftex 164 | )) 165 | \Gin@req@height=\dimen120 166 | \Gin@req@width=\dimen121 167 | ) 168 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/tools\multicol.sty" 169 | Package: multicol 2018/12/27 v1.8v multicolumn formatting (FMi) 170 | \c@tracingmulticols=\count103 171 | \mult@box=\box29 172 | \multicol@leftmargin=\dimen122 173 | \c@unbalance=\count104 174 | \c@collectmore=\count105 175 | \doublecol@number=\count106 176 | \multicoltolerance=\count107 177 | \multicolpretolerance=\count108 178 | \full@width=\dimen123 179 | \page@free=\dimen124 180 | \premulticols=\dimen125 181 | \postmulticols=\dimen126 182 | \multicolsep=\skip49 183 | \multicolbaselineskip=\skip50 184 | \partial@page=\box30 185 | \last@line=\box31 186 | \maxbalancingoverflow=\dimen127 187 | \mult@rightbox=\box32 188 | \mult@grightbox=\box33 189 | \mult@gfirstbox=\box34 190 | \mult@firstbox=\box35 191 | \@tempa=\box36 192 | \@tempa=\box37 193 | \@tempa=\box38 194 | \@tempa=\box39 195 | \@tempa=\box40 196 | \@tempa=\box41 197 | \@tempa=\box42 198 | \@tempa=\box43 199 | \@tempa=\box44 200 | \@tempa=\box45 201 | \@tempa=\box46 202 | \@tempa=\box47 203 | \@tempa=\box48 204 | \@tempa=\box49 205 | \@tempa=\box50 206 | \@tempa=\box51 207 | \@tempa=\box52 208 | \c@columnbadness=\count109 209 | \c@finalcolumnbadness=\count110 210 | \last@try=\dimen128 211 | \multicolovershoot=\dimen129 212 | \multicolundershoot=\dimen130 213 | \mult@nat@firstbox=\box53 214 | \colbreak@box=\box54 215 | \mc@col@check@num=\count111 216 | ) ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/tools\array.sty" 217 | Package: array 2018/12/30 v2.4k Tabular extension package (FMi) 218 | \col@sep=\dimen131 219 | \ar@mcellbox=\box55 220 | \extrarowheight=\dimen132 221 | \NC@list=\toks27 222 | \extratabsurround=\skip51 223 | \backup@length=\skip52 224 | \ar@cellbox=\box56 225 | ) (ProblemSet1.aux) 226 | \openout1 = `ProblemSet1.aux'. 227 | 228 | LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 14. 229 | LaTeX Font Info: ... okay on input line 14. 230 | LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 14. 231 | LaTeX Font Info: ... okay on input line 14. 232 | LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 14. 233 | LaTeX Font Info: ... okay on input line 14. 234 | LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 14. 235 | LaTeX Font Info: ... okay on input line 14. 236 | LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 14. 237 | LaTeX Font Info: ... okay on input line 14. 238 | LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 14. 239 | LaTeX Font Info: ... okay on input line 14. 240 | 241 | *geometry* driver: auto-detecting 242 | *geometry* detected driver: pdftex 243 | *geometry* verbose mode - [ preamble ] result: 244 | * driver: pdftex 245 | * paper: 246 | * layout: 247 | * layoutoffset:(h,v)=(0.0pt,0.0pt) 248 | * modes: 249 | * h-part:(L,W,R)=(72.26999pt, 469.75502pt, 72.26999pt) 250 | * v-part:(T,H,B)=(72.26999pt, 650.43001pt, 72.26999pt) 251 | * \paperwidth=614.295pt 252 | * \paperheight=794.96999pt 253 | * \textwidth=469.75502pt 254 | * \textheight=650.43001pt 255 | * \oddsidemargin=0.0pt 256 | * \evensidemargin=0.0pt 257 | * \topmargin=-37.0pt 258 | * \headheight=12.0pt 259 | * \headsep=25.0pt 260 | * \topskip=10.0pt 261 | * \footskip=30.0pt 262 | * \marginparwidth=65.0pt 263 | * \marginparsep=11.0pt 264 | * \columnsep=10.0pt 265 | * \skip\footins=9.0pt plus 4.0pt minus 2.0pt 266 | * \hoffset=0.0pt 267 | * \voffset=0.0pt 268 | * \mag=1000 269 | * \@twocolumnfalse 270 | * \@twosidefalse 271 | * \@mparswitchfalse 272 | * \@reversemarginfalse 273 | * (1in=72.27pt=25.4mm, 1cm=28.453pt) 274 | 275 | 276 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/context/base/mkii\supp-pd 277 | f.mkii" 278 | [Loading MPS to PDF converter (version 2006.09.02).] 279 | \scratchcounter=\count112 280 | \scratchdimen=\dimen133 281 | \scratchbox=\box57 282 | \nofMPsegments=\count113 283 | \nofMParguments=\count114 284 | \everyMPshowfont=\toks28 285 | \MPscratchCnt=\count115 286 | \MPscratchDim=\dimen134 287 | \MPnumerator=\count116 288 | \makeMPintoPDFobject=\count117 289 | \everyMPtoPDFconversion=\toks29 290 | ) 291 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/oberdiek\epstopdf-b 292 | ase.sty" 293 | Package: epstopdf-base 2016/05/15 v2.6 Base part for package epstopdf 294 | 295 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\infwarer 296 | r.sty" 297 | Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) 298 | ) 299 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/oberdiek\grfext.sty 300 | " 301 | Package: grfext 2016/05/16 v1.2 Manage graphics extensions (HO) 302 | 303 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\kvdefine 304 | keys.sty" 305 | Package: kvdefinekeys 2016/05/16 v1.4 Define keys (HO) 306 | 307 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\ltxcmds. 308 | sty" 309 | Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) 310 | ))) 311 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/oberdiek\kvoptions. 312 | sty" 313 | Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) 314 | 315 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\kvsetkey 316 | s.sty" 317 | Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) 318 | 319 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\etexcmds 320 | .sty" 321 | Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) 322 | 323 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\ifluatex 324 | .sty" 325 | Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) 326 | Package ifluatex Info: LuaTeX not detected. 327 | )))) 328 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\pdftexcm 329 | ds.sty" 330 | Package: pdftexcmds 2019/07/25 v0.30 Utility functions of pdfTeX for LuaTeX (HO 331 | ) 332 | Package pdftexcmds Info: LuaTeX not detected. 333 | Package pdftexcmds Info: \pdf@primitive is available. 334 | Package pdftexcmds Info: \pdf@ifprimitive is available. 335 | Package pdftexcmds Info: \pdfdraftmode found. 336 | ) 337 | Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 338 | 38. 339 | Package grfext Info: Graphics extension search list: 340 | (grfext) [.pdf,.png,.jpg,.mps,.jpeg,.jbig2,.jb2,.PDF,.PNG,.JPG,.JPE 341 | G,.JBIG2,.JB2,.eps] 342 | (grfext) \AppendGraphicsExtensions on input line 456. 343 | ) 344 | LaTeX Font Info: Try loading font information for U+msa on input line 17. 345 | 346 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsfonts\umsa.fd" 347 | File: umsa.fd 2013/01/14 v3.01 AMS symbols A 348 | ) 349 | LaTeX Font Info: Try loading font information for U+msb on input line 17. 350 | 351 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsfonts\umsb.fd" 352 | File: umsb.fd 2013/01/14 v3.01 AMS symbols B 353 | ) 354 | Underfull \hbox (badness 10000) in paragraph at lines 20--22 355 | 356 | [] 357 | 358 | 359 | Underfull \hbox (badness 10000) in paragraph at lines 36--37 360 | 361 | [] 362 | 363 | 364 | Underfull \hbox (badness 10000) in paragraph at lines 40--41 365 | 366 | [] 367 | 368 | 369 | Underfull \hbox (badness 10000) in paragraph at lines 44--46 370 | 371 | [] 372 | 373 | [1 374 | 375 | {C:/Users/User/AppData/Local/MiKTeX/2.9/pdftex/config/pdftex.map}] 376 | Underfull \hbox (badness 10000) in paragraph at lines 60--61 377 | 378 | [] 379 | 380 | 381 | Underfull \hbox (badness 10000) in paragraph at lines 65--66 382 | 383 | [] 384 | 385 | 386 | Underfull \hbox (badness 10000) in paragraph at lines 70--72 387 | 388 | [] 389 | 390 | 391 | Underfull \hbox (badness 10000) in paragraph at lines 76--77 392 | 393 | [] 394 | 395 | 396 | Underfull \hbox (badness 10000) in paragraph at lines 88--89 397 | 398 | [] 399 | 400 | 401 | Underfull \hbox (badness 10000) in paragraph at lines 89--90 402 | 403 | [] 404 | 405 | 406 | Underfull \hbox (badness 10000) in paragraph at lines 91--92 407 | 408 | [] 409 | 410 | 411 | Underfull \hbox (badness 10000) in paragraph at lines 93--94 412 | 413 | [] 414 | 415 | 416 | Underfull \hbox (badness 10000) in paragraph at lines 96--97 417 | 418 | [] 419 | 420 | [2] 421 | Underfull \hbox (badness 10000) in paragraph at lines 104--105 422 | 423 | [] 424 | 425 | 426 | Underfull \hbox (badness 10000) in paragraph at lines 106--107 427 | 428 | [] 429 | 430 | 431 | Underfull \hbox (badness 10000) in paragraph at lines 110--111 432 | 433 | [] 434 | 435 | 436 | Underfull \hbox (badness 10000) in paragraph at lines 113--114 437 | 438 | [] 439 | 440 | 441 | Underfull \hbox (badness 10000) in paragraph at lines 115--116 442 | 443 | [] 444 | 445 | 446 | Underfull \hbox (badness 10000) in paragraph at lines 119--120 447 | 448 | [] 449 | 450 | 451 | Underfull \hbox (badness 10000) in paragraph at lines 121--122 452 | 453 | [] 454 | 455 | 456 | Underfull \hbox (badness 10000) in paragraph at lines 126--128 457 | 458 | [] 459 | 460 | 461 | Underfull \hbox (badness 10000) in paragraph at lines 130--131 462 | 463 | [] 464 | 465 | 466 | Underfull \hbox (badness 10000) in paragraph at lines 132--133 467 | 468 | [] 469 | 470 | 471 | Underfull \hbox (badness 10000) in paragraph at lines 136--137 472 | 473 | [] 474 | 475 | 476 | Underfull \hbox (badness 10000) in paragraph at lines 140--141 477 | 478 | [] 479 | 480 | [3] 481 | Underfull \hbox (badness 10000) in paragraph at lines 143--144 482 | 483 | [] 484 | 485 | 486 | Underfull \hbox (badness 10000) in paragraph at lines 147--148 487 | 488 | [] 489 | 490 | 491 | Underfull \hbox (badness 10000) in paragraph at lines 149--150 492 | 493 | [] 494 | 495 | 496 | Underfull \hbox (badness 10000) in paragraph at lines 152--153 497 | 498 | [] 499 | 500 | 501 | Underfull \hbox (badness 10000) in paragraph at lines 155--156 502 | 503 | [] 504 | 505 | 506 | Underfull \hbox (badness 10000) in paragraph at lines 162--163 507 | 508 | [] 509 | 510 | 511 | Underfull \hbox (badness 10000) in paragraph at lines 164--165 512 | 513 | [] 514 | 515 | 516 | Underfull \hbox (badness 10000) in paragraph at lines 168--169 517 | 518 | [] 519 | 520 | 521 | Underfull \hbox (badness 10000) in paragraph at lines 176--177 522 | 523 | [] 524 | 525 | [4] 526 | Underfull \hbox (badness 10000) in paragraph at lines 183--184 527 | 528 | [] 529 | 530 | 531 | Underfull \hbox (badness 10000) in paragraph at lines 185--186 532 | 533 | [] 534 | 535 | 536 | Underfull \hbox (badness 10000) in paragraph at lines 189--190 537 | 538 | [] 539 | 540 | 541 | Underfull \hbox (badness 10000) in paragraph at lines 196--197 542 | 543 | [] 544 | 545 | 546 | Underfull \hbox (badness 10000) in paragraph at lines 198--199 547 | 548 | [] 549 | 550 | 551 | Overfull \hbox (132.18982pt too wide) detected at line 209 552 | \OT1/cmr/m/n/10 = [][]\OML/cmm/m/it/10 ; [][] 553 | [] 554 | 555 | 556 | Underfull \hbox (badness 10000) in paragraph at lines 217--218 557 | 558 | [] 559 | 560 | [5] [6] (ProblemSet1.aux) ) 561 | Here is how much of TeX's memory you used: 562 | 3061 strings out of 492895 563 | 42761 string characters out of 3124202 564 | 126182 words of memory out of 3000000 565 | 6930 multiletter control sequences out of 15000+200000 566 | 10591 words of font info for 42 fonts, out of 3000000 for 9000 567 | 1141 hyphenation exceptions out of 8191 568 | 41i,11n,32p,410b,199s stack positions out of 5000i,500n,10000p,200000b,50000s 569 | 587 | Output written on ProblemSet1.pdf (6 pages, 165327 bytes). 588 | PDF statistics: 589 | 81 PDF objects out of 1000 (max. 8388607) 590 | 0 named destinations out of 1000 (max. 500000) 591 | 1 words of extra memory for PDF output out of 10000 (max. 10000000) 592 | 593 | -------------------------------------------------------------------------------- /PS1/ProblemSet1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS1/ProblemSet1.pdf -------------------------------------------------------------------------------- /PS1/ProblemSet1.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS1/ProblemSet1.synctex.gz -------------------------------------------------------------------------------- /PS1/ProblemSet1.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt]{article} 2 | 3 | \usepackage[margin=1in]{geometry} 4 | \usepackage{amsmath,amsthm,amssymb, graphicx, multicol, array} 5 | 6 | \newcommand{\N}{\mathbb{N}} 7 | \newcommand{\Z}{\mathbb{Z}} 8 | 9 | \newenvironment{problem}[2][Problem]{\begin{trivlist} 10 | \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} 11 | 12 | 13 | 14 | \begin{document} 15 | \title{Problem Set 1} 16 | \author{Hoang Nguyen, Huy Nguyen} 17 | \maketitle 18 | 19 | \begin{problem}{1} 20 | \item 1. 21 | We have:\\ 22 | \[\mathbb{E}[X_n]=(1-\frac{1}{n^2}).\frac{1}{n}+\frac{1}{n^2}.n\] 23 | \[=\frac{1}{n}-\frac{1}{n^3}+\frac{1}{n}\] 24 | \[=\frac{2}{n}-\frac{1}{n^3}\] 25 | 26 | On the other hand: 27 | \[lim_{n \rightarrow \infty}\mathbb{P}[|X_n| > \epsilon]=lim_{n \rightarrow \infty}\mathbb{P}[X_n > \epsilon]\leq \frac{lim_{n \rightarrow \infty}\mathbb{E}[X_n]}{\epsilon}=\frac{lim_{n \rightarrow \infty}(\frac{2}{n}-\frac{1}{n^3})}{\epsilon}=0\] 28 | 29 | 30 | Hence $X_n$ converge in probability 31 | 32 | \[\mathbb{E}[X_n^2]=(1-\frac{1}{n^2}).\frac{1}{n^2}+\frac{1}{n^2}.n^2\] 33 | \[=\frac{1}{n^2}-\frac{1}{n^4}+1\] 34 | \[\Rightarrow lim_{n \rightarrow \infty}\mathbb{E}[X_n^2]=1\] 35 | 36 | We have: \\ 37 | \[lim_{n \rightarrow \infty}\mathbb{E}[(X_n-X)^2]=lim_{n \rightarrow \infty}(\mathbb{E}[X_n^2]-2\mathbb{E}[X_n]\mathbb{E}[X]+\mathbb{E}[X^2])\] 38 | \[=1+\mathbb{E}[X^2]\qquad (because\quad \mathbb{E}[X_n]=0)\] 39 | 40 | If $X_n$ converge in $\mathbb{L}^2$ $\Rightarrow$ $1+\mathbb{E}[X^2]=0 \Rightarrow \mathbb{E}[X^2]=-1 \quad (imposible)$\\ 41 | 42 | Hence, $X_n$ does not converge in $\mathbb{L}^2$ 43 | 44 | \item 2. 45 | We have:\\ 46 | \[\mathbb{E}[X_n]=p\] 47 | \[\mathbb{V}[X_n]=\frac{p(1-p)}{n}\] 48 | 49 | In addition: 50 | 51 | \[n\bar{X_n}=\sum_{i=1}^{n} X_i\] 52 | \[\Rightarrow \frac{n\bar{X_n}-\mu_n}{\sigma_n}=\frac{\sum_{i=1}^{n} X_i-p}{\frac{p(1-p)}{n}}\sim\mathbb{N}(0,1)\qquad (Central \quad Limit \quad Theorem)\] 53 | \[\Rightarrow \mathbb{P}[\frac{n\bar{X_n}-p}{\frac{p(1-p)}{n}}\lneq t]=\Phi(t)\] 54 | \[\Rightarrow \mathbb{P}\Big[ n\bar{X_n} \lneq \frac{tp(1-p)}{n}+p \Big]=\Phi(t)\] 55 | \[\Rightarrow \mathbb{P}[ n\bar{X_n} \lneq x]=\Phi(\frac{(x-p)n}{p(1-p)}\] 56 | \[\Rightarrow F_n(x)=\Phi(\frac{(x-p)n}{p(1-p)} \qquad (1)\] 57 | 58 | (1) is the CDF of distribution of n$\bar{X_n}$ 59 | 60 | We have:\\ 61 | \[\mathbb{E}[(\bar{X_n}-p)^2]= \mathbb{E}[\bar{X_n}^2]-2p\mathbb{E}[\bar{X_n}]+ p^2\] 62 | \[=\sigma_n+ (\mathbb{E}[\bar{X_n}])^2-2p\mathbb{E}[\bar{X_n}]+ p^2\] 63 | \[=\frac{p(1-p)}{n}+p^2-2p^2+ p^2\] 64 | \[=\frac{p(1-p)}{n}\] 65 | \[\Rightarrow lim_{n \rightarrow \infty} \mathbb{E}[(\bar{X_n}-p)^2]=0\]\\ 66 | 67 | $\Rightarrow \bar{X_n}$ converges to p in L 68 | 69 | 70 | \item 3. 71 | a) From Markov's Inequality: \\ 72 | \[\mathbb{P}[|X_n|> \epsilon]= \mathbb{P}[X_n>\epsilon] < \frac{\mathbb{E}[X_n]}{\epsilon}=\frac{\lambda}{\epsilon}= \frac{1}{n\epsilon}\] 73 | \[\Rightarrow \lim_{n \rightarrow \infty} \mathbb{P}[|X_n| > \epsilon]=0\] 74 | \[ \Rightarrow X_n \xrightarrow{\mathbb{P}}0\] 75 | 76 | b) We have: \\ 77 | \[\mathbb{P}[|nX_n|< \epsilon]= \mathbb{P}[X_n<\frac{\epsilon}{n}]=\sum_{i=0}^{\frac{\epsilon}{n}}e^{-\lambda}\frac{\lambda^i}{i!}\] 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | \end{problem} 86 | 87 | \begin{problem}{2} 88 | \item 1. True \\ 89 | \item 2.True \\ 90 | \item 3. 91 | \item 4.\\ 92 | 93 | We get the formula for the interval confident of Bernulli random variables:\\ 94 | \[\mathbb{L}=\Big[\bar{X_n}- \frac{1.96\sqrt{p(1-p)}}{\sqrt{n}},\bar{X_n}+ \frac{1.96\sqrt{p(1-p)}}{\sqrt{n}} \Big]\] 95 | 96 | In addition:\\ 97 | \[p=0.43\] 98 | \[n=100\] 99 | \[\Rightarrow \mathbb{L}=[0.33,0.53]\] 100 | 101 | \end{problem} 102 | 103 | \begin{problem}{3} 104 | \item 1. \\ 105 | 106 | We have:\\ 107 | \[\mu_n=\mathbb{E}_n[\bar{X_n}]=p\] 108 | \[\sigma_n=\frac{p(1-p)}{n}\] 109 | 110 | By using Central Limit Theorem:\\ 111 | \[\Rightarrow \bar{X_n}\sim \mathbb{N}(\mu,\frac{\sigma_n^2}{n} )\] 112 | \[\sqrt{n}\frac{\bar{X_n}-p}{\sqrt{p(1-p)}}\sim \mathbb{N}(0,1)\] 113 | \item 2. \\ 114 | 115 | We have: $Z \backsim N(0,1)$\\ 116 | 117 | Hence, $\mathbb{P}[|Z| \leqslant t]=\mathbb{P}[-t\leqslant Z \leqslant t]= \phi(t)-\phi(-t)= \phi(t)-(1-\phi(t))=2\phi(t)-1= 2\mathbb{P}[Z \leqslant t]-1$. 118 | 119 | \item 3. \\ 120 | 121 | We have:\\ 122 | \[\mathbb{P}[p \in I_t]=\mathbb{P}\Big[\bar{X_n}- \frac{t\sqrt{p(1-p)}}{\sqrt{n}}\leqslant p \leqslant\bar{X_n}+ \frac{t\sqrt{p(1-p)}}{\sqrt{n}} \Big]\] 123 | \[=\mathbb{P}\Big[ \Big|\sqrt{n}\frac{\bar{X_n}-p}{\sqrt{p(1-p)}}\Big|\leqslant t\Big]\] 124 | Because: 125 | \[\sqrt{n}\frac{\bar{X_n}-p}{\sqrt{p(1-p)}}\sim \mathbb{N}(0,1)\] 126 | \[\mathbb{P}[|Z| \leqslant t]=2\mathbb{P}[Z \leqslant t]-1\quad (Z=\mathbb{N}(0,1))\] 127 | Hence, \\ 128 | \[\mathbb{P}[p \in I_t]\rightarrow 2\phi(t)-1, n\rightarrow\infty \quad (using\quad the \quad previous \quad questions)\] 129 | 130 | \item 4. \\ 131 | 132 | We have:\\ 133 | \[\mathbb{P}[p \in I_t]\rightarrow 2\phi(t)-1=0.95\] 134 | \[\Rightarrow {t_0}=1.96\] 135 | 136 | Hence, \\ 137 | \[I_{t_0}=\Big[\bar{X_n}- \frac{t\sqrt{p(1-p)}}{\sqrt{n}}, \bar{X_n}+ \frac{t\sqrt{p(1-p)}}{\sqrt{n}} \Big]\] 138 | \[=\Big[\bar{X_n}- \frac{1.96\sqrt{p(1-p)}}{\sqrt{n}}, \bar{X_n}+ \frac{1.96\sqrt{p(1-p)}}{\sqrt{n}} \Big]\] 139 | 140 | Using the fact that: \\ 141 | \[0 \leqslant\sqrt{p(1-p)} \leqslant \frac{1}{4}\] 142 | 143 | We get: \\ 144 | \[I_t=\Big[ \Big]\] 145 | \[\Rightarrow I_t=\Big[ \bar{X_n}-\frac{0.49}{\sqrt{n}},\bar{X_n}+\frac{0.49}{\sqrt{n}} \Big]\] 146 | 147 | \item 5. \\ 148 | 149 | a) Using Cauchy's inequality: \\ 150 | \[p(1-p) \leqslant \frac{q+1-p}{4}=\frac{1}{4}\] 151 | 152 | b)From the previous question: \\ 153 | \[0 \leqslant\sqrt{p(1-p)} \leqslant \frac{1}{2}\] 154 | 155 | We get: \\ 156 | \[I_t=\Big[ \Big]\] 157 | \[\Rightarrow I_t=\Big[ \bar{X_n}-\frac{0.98}{\sqrt{n}},\bar{X_n}+\frac{0.98}{\sqrt{n}} \Big]\] 158 | 159 | c) Because $p \leqslant 0.3 \Rightarrow p(1-p) \leqslant 0.21$ 160 | \[\Rightarrow J_1=\Big[ \bar{X_n}-\frac{1.96\sqrt{0.21}}{\sqrt{n}},\bar{X_n}+\frac{1.96\sqrt{02.1}}{\sqrt{n}} \Big]= \Big[ \bar{X_n}-\frac{0.89}{\sqrt{n}},\bar{X_n}+\frac{0.89}{\sqrt{n}} \Big]\] 161 | 162 | \item 6. \\ 163 | 164 | a) We have:\\ 165 | \[ p \in I_{t_0} \Longleftrightarrow \bar{X_n}-\frac{t_0\sqrt{p(1-p)}}{\sqrt{n}}\leqslant p \leqslant\bar{X_n}+\frac{t_0\sqrt{p(1-p)}}{\sqrt{n}} \] 166 | \[\Longleftrightarrow p^2(1+\frac{{t_0}^2}{n})-(2\bar{X_n}+\frac{{t_0}^2}{n})p+{\bar{X_n}}^2 \leqslant 0\] 167 | 168 | b) Solving this inequality we get the value of p in form:\\ 169 | \[\Big[\frac{-b-\sqrt{\Delta}}{2a},\frac{-b+\sqrt{\Delta}}{2a}\Big] \] 170 | 171 | With: 172 | \[b=-(2\bar{X_n}+\frac{{t_0}^2}{n})\] 173 | \[\Delta=\frac{{t_0}^4}{n^2}+4\bar{X_n}\frac{{t_0}^2}{n}-\frac{4\bar{X_n}^2{t_0}^2}{n}\] 174 | \[a=1+\frac{{t_0}^2}{n}\] 175 | 176 | c) From the previous question, we have : \\ 177 | 178 | \[J_2= \Big[\frac{2\bar{X_n}+\frac{{t_0}^2}{n}-\sqrt{\frac{{t_0}^4}{n^2}+4\bar{X_n}\frac{{t_0}^2}{n}-\frac{4\bar{X_n}^2{t_0}^2}{n}}}{2+\frac{2{t_0}^2}{n}}, \frac{2\bar{X_n}+\frac{{t_0}^2}{n}+\sqrt{\frac{{t_0}^4}{n^2}+4\bar{X_n}\frac{{t_0}^2}{n}-\frac{4\bar{X_n}^2{t_0}^2}{n}}}{2+\frac{2{t_0}^2}{n}}\Big]\] 179 | 180 | 181 | 182 | 183 | \item 7. \\ 184 | 185 | Replace p with $\bar{X_n}$, we get: \\ 186 | 187 | \[J_3=\Big[\bar{X_n}- \frac{{t_0}\sqrt{\bar{X_n}(1-\bar{X_n})}}{\sqrt{n}},\bar{X_n}+ \frac{{t_0}\sqrt{\bar{X_n}(1-\bar{X_n})}}{\sqrt{n}} \Big]\] 188 | 189 | As n goes to infinity, we have:\\ 190 | 191 | \[\bar{X_n} \xrightarrow{\mathbb{P}} \mu=p\] 192 | \[\Rightarrow J_3\longrightarrow \Big[\bar{X_n}- \frac{{t_0}\sqrt{p(1-p)}}{\sqrt{n}}, \bar{X_n}+ \frac{{t_0}\sqrt{p(1-p)}}{\sqrt{n}} \Big]\longrightarrow 0.95\] 193 | 194 | 195 | 196 | \item 8. \\ 197 | 198 | a) We have from the question: \\ 199 | \[n=10000\] 200 | \[\bar{X_n}=1-0.7341=0.2659\] 201 | 202 | 203 | 204 | 205 | \[J_1=\Big[ \bar{X_n}-\frac{0.89}{\sqrt{n}},\bar{X_n}+\frac{0.89}{\sqrt{n}} \Big]=\Big[ 0.2659-\frac{0.89}{\sqrt{100}}, 0.2659+\frac{0.89}{\sqrt{100}}\Big]=\Big[0.1769,0.3549 \Big]\] 206 | 207 | \[J_2= \Big[\frac{2\bar{X_n}+\frac{{t_0}^2}{n}-\sqrt{\frac{{t_0}^4}{n^2}+4\bar{X_n}\frac{{t_0}^2}{n}-\frac{4\bar{X_n}^2{t_0}^2}{n}}}{2+\frac{2{t_0}^2}{n}}, \frac{2\bar{X_n}+\frac{{t_0}^2}{n}+\sqrt{\frac{{t_0}^4}{n^2}+4\bar{X_n}\frac{{t_0}^2}{n}-\frac{4\bar{X_n}^2{t_0}^2}{n}}}{2+\frac{2{t_0}^2}{n}}\Big]\] 208 | 209 | \[=\Big[\frac{2\times0.2659+\frac{1.96^2}{100}-\sqrt{\frac{{1.96}^4}{100^2}+4\times 0.2659\times\frac{1.96^2}{100}-\frac{4\times0.2659^2\times 1.96^2}{100}}}{2+\frac{2\times 1.96^2}{100}}, \frac{2\times0.2659+\frac{1.96^2}{100}+\sqrt{\frac{{1.96}^4}{100^2}+4\times 0.2659\times\frac{1.96^2}{100}-\frac{4\times0.2659^2\times 1.96^2}{100}}}{2+\frac{2\times 1.96^2}{100}}\Big]\] 210 | \[=\Big[0.189,0.365\Big]\] 211 | 212 | 213 | \[J_3=\Big[\bar{X_n}- \frac{{t_0}\sqrt{\bar{X_n}(1-\bar{X_n})}}{\sqrt{n}},\bar{X_n}+ \frac{{t_0}\sqrt{\bar{X_n}(1-\bar{X_n})}}{\sqrt{n}} \Big]\] 214 | \[=\Big[0.2659- \frac{1.96\sqrt{0.2659\times(1-0.2659)}}{\sqrt{100}},0.2659+ \frac{1.96\sqrt{0.2659\times(1-0.2659)}}{\sqrt{100}}\Big]\] 215 | \[=\Big[0.1793,0.3524 \Big]\] 216 | 217 | b) From the previous question, we get the tightest interval of p is $J_3$, so we obtain $J_3$ to solve this problem. We have:\\ 218 | 219 | The length of the interval is at most 0.05: 220 | \[\Rightarrow \frac{{t_0}\sqrt{\bar{X_n}(1-\bar{X_n})}}{\bar{X_n}\sqrt{n}} \leqslant 0.025\] 221 | \[\Longleftrightarrow {\Big(\frac{{t_0}\sqrt{\bar{X_n}(1-\bar{X_n})}}{\bar{X_n}\times0.025} \Big)}^2 \leqslant n\] 222 | \[\Longleftrightarrow 16970 \leqslant n\] 223 | 224 | So, the minimal of n is 16970. 225 | 226 | 227 | 228 | 229 | \end{problem} 230 | \end{document} -------------------------------------------------------------------------------- /PS10/PropblemSet10.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 4 | \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined 5 | \global\let\oldcontentsline\contentsline 6 | \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} 7 | \global\let\oldnewlabel\newlabel 8 | \gdef\newlabel#1#2{\newlabelxx{#1}#2} 9 | \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} 10 | \AtEndDocument{\ifx\hyper@anchor\@undefined 11 | \let\contentsline\oldcontentsline 12 | \let\newlabel\oldnewlabel 13 | \fi} 14 | \fi} 15 | \global\let\hyper@last\relax 16 | \gdef\HyperFirstAtBeginDocument#1{#1} 17 | \providecommand\HyField@AuxAddToFields[1]{} 18 | \providecommand\HyField@AuxAddToCoFields[2]{} 19 | -------------------------------------------------------------------------------- /PS10/PropblemSet10.out: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS10/PropblemSet10.out -------------------------------------------------------------------------------- /PS10/PropblemSet10.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS10/PropblemSet10.pdf -------------------------------------------------------------------------------- /PS10/PropblemSet10.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS10/PropblemSet10.synctex.gz -------------------------------------------------------------------------------- /PS10/PropblemSet10.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt]{article} 2 | \usepackage{hyperref} 3 | \usepackage[margin=1in]{geometry} 4 | \usepackage{amsmath,amsthm,amssymb, graphicx, multicol, array} 5 | 6 | \newcommand{\N}{\mathbb{N}} 7 | \newcommand{\Z}{\mathbb{Z}} 8 | 9 | \newenvironment{problem}[2][Problem]{\begin{trivlist} 10 | \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} 11 | 12 | 13 | 14 | \begin{document} 15 | \title{Problem Set 10} 16 | \author{Hoang Nguyen, Huy Nguyen} 17 | \maketitle 18 | 19 | \begin{problem}{1} 20 | \item 1. 21 | We have this result from problem 2 in Problem Set 3: 22 | \[ l(\theta) = \sum_{i=1}^{n} \Big ( -\frac{1}{2} log(\theta) -\frac{1}{2\theta} X_i^2\Big ) \] 23 | Taking deravite ans set to equal 0, we get: 24 | \[\frac{\partial l(\theta)}{\partial\theta} = \sum_{i=1}^{n}(-\frac{1}{2\theta} + \frac{1}{2\theta^2} X_i^2)=0 \Longleftrightarrow \theta^{MLE} = \frac{\sum_{i=1}^{n} X_i^2}{n} \tag{1}\] 25 | 26 | 27 | 28 | 29 | \item 2. 30 | From (1), we have the second deraviate of log-likelihood: 31 | \[\frac{\partial l(\theta)}{\partial \theta^2} =\sum_{i=1}^{n} \Big ( \frac{1}{2\theta^2} - \frac{1}{2\theta^3}X_i^2 \Big ) \Leftrightarrow I(\theta) = -\sum_{i=1}^{n}\mathbb{E}[\frac{1}{2\theta^2} - \frac{1}{2\theta^3}X_i^2]= \frac{n}{2\theta^2}\] 32 | Hence, 33 | \[(\theta^{MLE} - \theta) \xrightarrow[n \rightarrow \infty]{\text{(d)}} N(0, \frac{2\theta^2}{n}) \] 34 | 35 | 36 | 37 | 38 | \item 3.\\ 39 | a) From previous question, we know: $I(\theta) = \frac{n}{2\theta^2}$. Hence, $\pi(\theta) = c\sqrt{det I(\theta)} = c\sqrt{\frac{n}{2\theta^2}}$. This is improper prior\\ 40 | b) We have 41 | \[ \pi(\theta| X_1,..,X_n) = \frac{\pi(\theta) p_n(X_1,..,X_n|\theta))}{\int_{0}^{\infty} p_n(X_1,..,X_n|t)d\pi(t)} = \frac{c\sqrt{\frac{n}{2\theta^2}} \frac{1}{(2\pi\theta)^{\frac{n}{2}}} e^{-\frac{1}{2\theta} \sum_{i}^{n} X_i^2}}{\int_{0}^{\infty} p_n(X_1,..,X_n|t)d\pi(t)} = \frac{c\sqrt{n}}{(2\pi\theta)^{\frac{n}{2}} \sqrt{2\theta^2}} e^{-\frac{1}{2\theta} \sum_{i}^{n} X_i^2}\] 42 | Hence, we can rewrite the posterio: 43 | \[\frac{c\sqrt{n}}{(2\pi\theta)^{\frac{n}{2}} \sqrt{2\theta^2}} e^{-\frac{1}{2\theta} \sum_{i}^{n} X_i^2} = \frac{c\sqrt{n}}{(2\pi)^{\frac{n}{2}} \sqrt{2}} \frac{e^{-\frac{\frac{1}{2} \sum_{i=1}^{n}}{\theta}}}{\theta^{\frac{n}{2} +1}} = \frac{c\sqrt{n}}{(2\pi)^{\frac{n}{2}} \sqrt{2}} \frac{e^{-\frac{\beta}{\theta}}}{\theta^{\alpha+1}} \] 44 | Where $\alpha = \frac{n}{2}$, $\beta = \frac{\sum_{i=1}^{n} X_i^2}{2}$ and $c$ is the constant satisfied $\frac{c\sqrt{n}}{(2\pi)^{\frac{n}{2}} \sqrt{2}} = \frac{\beta^{\alpha}}{\tau(\alpha)}$. This posterior is Gamma distribution with parameters $\alpha$ and $\beta$. 45 | c)\\ 46 | We know $\hat{\theta^{(\pi)}}= \int_{0}^{\infty} \theta d \pi(\theta| X_1,..,X_n)$ is the expectation of Gamma distribution with parameters $\alpha$ and $\beta$. Hence, $\hat{\theta^{(\pi)}} = \frac{\beta}{\alpha - 1} = \frac{\sum_{i=1}^{n} X_i^2}{n -2}$. 47 | 48 | 49 | \end{problem} 50 | 51 | \begin{problem}{2} 52 | \item 1. Conditionally on $\beta$, we have $Y-X\beta \sim N_p(0, \sigma^2 I_n)$. Hence, $Y \sim N_p(X\beta, \sigma^2 I_n)$. 53 | \item 2.\\ 54 | a) We know: 55 | \[ \pi(\beta| X_1,..,X_n) \propto \pi(\beta) p_n(X_1,..,X_n|\beta)) \propto e^{\frac{1}{\sigma^2} ||Y - X\beta||_2^{2}} e^{\frac{1}{\tau^2}||\beta||_2^2} = e^{\frac{1}{\sigma^2} \Big ( ||Y - X\beta||_2^{2} + \frac{\sigma^2}{\tau^2} ||\beta||_2^2 \Big ) } \] 56 | b) Let $\pi(\beta| X_1,..,X_n) = Ke^{\frac{1}{\sigma^2} \Big ( ||Y - X\beta||_2^{2} + \frac{\sigma^2}{\tau^2} ||\beta||_2^2 \Big ) } $\\ 57 | We know: 58 | \[\frac{1}{\sigma^2} \Big ( ||Y - X\beta||_2^{2} + \frac{\sigma^2}{\tau^2} ||\beta||_2^2 \Big ) = \frac{1}{\sigma^2}\Big ( (Y - X\beta)^{T}(Y - X\beta) + \frac{\sigma^2}{\tau^2} \beta^{T}\beta \Big )\] 59 | \[ = \frac{1}{\sigma^2}\Big ( Y^{T}Y - 2\beta^{T}X^{T}Y + \beta^{T}X^{T}X\beta + \frac{\sigma^2}{\tau^2} \beta^{T}\beta \Big ) = \frac{1}{\sigma^2}\Big ( Y^{T}Y - 2\beta^{T}X^{T}Y \Big )+ \beta^{T}(\frac{1}{\sigma^2} X^{T}X + \frac{1}{\tau^2} I)\beta \] 60 | Hence, 61 | \[ \pi(\beta| X_1,..,X_n) = Ke^{\frac{1}{\sigma^2}\Big ( Y^{T}Y - 2\beta^{T}X^{T}Y\Big ) + \beta^{T}(\frac{1}{\sigma^2} X^{T}X + \frac{1}{\tau^2} I)\beta} = K_1e^{\frac{-2}{\sigma^2}\beta^{T}X^{T}Y + \beta^{T}(\frac{1}{\sigma^2} X^{T}X + \frac{1}{\tau^2} I)\beta }\] 62 | \[= K_1e^{ \beta^{T}(\frac{1}{\sigma^2} X^{T}X + \frac{1}{\tau^2} I)\beta -\frac{2}{\sigma^2}\beta^{T}X^{T}Y }\] 63 | Let $\sum^{-1} = \frac{1}{\sigma^2} X^{T}X + \frac{1}{\tau^2} I$ and $\mu = (X^{T}X + \frac{\sigma^2}{\tau^2} I)^{-1}X^{T}y$. We can see that $\frac{1}{\sigma^2} X^{T}y = \sum^{-1}\mu$. Thereforce, we can rewrite 64 | \[ \pi(\beta| X_1,..,X_n) = K_1 e^{\beta^T \sum^{-1} \beta - 2\beta \sum^{-1}\mu } = K_1 e^{\beta^T \sum^{-1} \beta - 2\beta \sum^{-1}\mu }= K_1 e^{\beta^T \sum^{-1} \beta - 2\beta \sum^{-1}\mu + \mu^T\sum^{-1}\mu - \mu^T\sum^{-1}\mu}\] 65 | We see that $ \mu^T\sum^{-1}\mu$ depends on $X$ and $Y$ which are constants, hence we can get rid this constant by rewrite: 66 | \[\pi(\beta| X_1,..,X_n) = K_2 e^{\beta^T \sum^{-1} \beta - 2\beta \sum^{-1}\mu + \mu^T\sum^{-1}\mu} = K_2e^{(\beta - \mu)^{T}\sum^{-1}(\beta - \mu)}\] 67 | c) The posterior mean of $\beta$ is the expectation of $g(\beta)$ which is $\mu = (X^{T}X + \frac{\sigma^2}{\tau^2} I)^{-1}X^{T}y$ 68 | \item 3. \\ 69 | a) Consider function $f(t)= ||Y-Xt||^2 + \lambda ||t||^2$. We take derivate and set it to be zero: 70 | \[\frac{\partial f(t)}{\partial t} = 2X^{T}Xt - 2X^{T}Y + 2\lambda = 0 \Longleftrightarrow \hat{\beta} = t = (X^{T}X + \lambda I)^{-1}X^{T}Y\] 71 | b) We know the posterio mean is $\mu = (X^{T}X + \frac{\sigma^2}{\tau^2} I)^{-1}X^{T}y$. Hence, there exists a $\tau^2$ such that $\frac{\sigma^2}{\tau^2} = \lambda\Longleftrightarrow \tau^2 = \frac{\sigma^2}{\lambda}$.\\ 72 | c) We know $Y \sim N_{p}(X\beta, \sigma^2 I)$. Hence 73 | \[\hat{\beta} = (X^{T}X + \lambda I)^{-1}X^{T}Y \sim N_{p} Big (0, \sigma^2\Big ((X^{T}X + \lambda I)^{-1}X^{T} \Big )^{T}\Big ((X^{T}X + \lambda I)^{-1}X^{T} \Big ) \Big ) = N_{p}(0, \sigma^2(X^{T}X + \lambda I)^{-1}) \] 74 | d) We have $\mathbb{E}[ ||\hat{\beta} - \beta||^2_2]=\sum_{i=1}^{D}\mathbb{E}[(\hat{\beta}_i -\beta_i)^2]=\sum_{i=1}^{D}Var(\hat{\beta}_i)=\sigma^2 tr((X^{T}X + \lambda I)^{-1})$.\\ 75 | 76 | \end{problem} 77 | \begin{problem}{3} 78 | \item 1. Covariance matrix is a square matrix giving the covariance between each pair of elements of a given random vector. In the matrix diagonal there are variances, i.e., the covariance of each element with itself. The dimension is $p$ by $p$. 79 | \item 2. The sample covariance matrix is the matrix whose elements are pairwise covariances of the vectors. 80 | \item 3. We have covariance matrix $C = \mathbb{E}[(X - \bar{X})(X-\bar{X})^{T}]$. Hence 81 | \[ u^{T}Cu = u^{T}\mathbb{E}[(X - \bar{X})(X-\bar{X})^{T}]u = \mathbb{E}[u^{T}(X - \bar{X})(X-\bar{X})^{T}u] = \mathbb{E}[\sigma^2]=\sigma^2 \geqslant 0\] 82 | \item 4. \\ 83 | We know $S=\frac{1}{n}\sum_{i=1}^{n}X_{i}X_{i}^{T} - \bar{X}\bar{X}^{T}$. Hence: 84 | \[u^{T}Su = \frac{1}{n}\sum_{i=1}^{n}u^{T}X_{i} (X_{i}^{T}u) - u\bar{X}(\bar{X}^{T}u) = \frac{1}{n}\sum_{i=1}^{n}(u^{T}X_{i})^2 - (u^{T}\bar{X})^2 \] 85 | On the other hand, $u^{T}\bar{X} = \frac{1}{n}\sum_{i=1}^{n}u^TX_{i}=\bar{u^{T}X}$. Hence $u^{T}Su = \frac{1}{n}\sum_{i=1}^{n}u^{T}X_{i} (X_{i}^{T}u) - \bar{u^{T}X}^2 \Rightarrow $ $u^{T}Su $ is sample variance of $u^{X}$, hecne this has to be greater or equal to 0. 86 | \item 5. \\ 87 | a) Let $B = AX$ and $b_{i}$ be the element ith of B. We have the element on i-row, j-column of covarinace maxtrix $B'$ of B is calculated: 88 | \[ B'_{ij} = Cov(B_{i}, B_{j}) = Cov(A_{i}X, A_{j}X) = \mathbb{E}[(A_{i}X)(A_{j}X)^{T}]) - \mathbb{E}[A_{i}X]\mathbb{E}[A_{j}X]^{T} = A_{i}\mathbb{E}[XX^{T}]A_{j}^{T} - A_{i}\mathbb{E}[X]\mathbb{E}[X]^{T}A_{j}^{T} \] 89 | \[ = A_{i}(\mathbb{E}[XX^{T}] -\mathbb{E}[X]\mathbb{E}[X]^{T} )A_{j}^{T} = A_{i}\sum A_{j}^{T} \] 90 | where $A_i$ denotes whole j row of matrix A. Hence, the covariance of $AX$ is $A\sum A^{T}$.\\ 91 | b) For any non-zero vector $ u \in \mathbb{R}^{q}$, we have: 92 | \[ u^{T}A\sum A^{T}u^{T} = (A^{T}u)\sum (A^{T}u)^{T} \] 93 | If $A^{T}$ has full column rank which means for any non-zero u, $A^{T}u = 0$ is impossible, hence $u^{T}A\sum A^{T}u^{T} = (A^{T}u)\sum (A^{T}u)^{T} > 0 \Rightarrow $ all eigenvalues of $A\sum A^{T}$ are positive, hence covariance of $AX$ is invertible. \\ 94 | If $A^{T}$ does not have full column rank, there exsists non-zero vector u such that $A^{T}u = 0$. That u make $u^{T}A\sum A^{T}u^{T} = (A^{T}u)\sum (A^{T}u)^{T} = 0$ which means there is at least an eigenvalue of $A\sum A^{T}$ that is equal to 0, hence covariance of $AX$ is not invertible.\\ 95 | c) We have $\sum = \mathbb{E}[(u^{T}X) (X^{T}u)] - \mathbb{E}[u^{T}X]\mathbb{E}[u^{T}X]^{T}= u^{T}\mathbb{E}[XX^{T}]u - u^{T}\mathbb{E}[X]\mathbb{E}[X]^{T}u = u^{T}\sum_{X}u$. 96 | \item 6.\\ 97 | a) Using similar methhof from previous questions, sample covariance matrix of $BX_1, BX_2,..,BX_n$ are $B\hat{\sum}_{1}B^{T}, B\hat{\sum}_{2}B^{T},...,B\hat{\sum}_{n}B^{T}$.\\ 98 | b) Using similar methhof from previous questions, sample covariance matrix of $u^{T}X_1, u^{T}X_2,..,u^{T}X_n$ are $u^{T}\hat{\sum}_{1}u, u^{T}\hat{\sum}_{2}u,...,u^{T}\hat{\sum}_{n}u$.\\ 99 | \item 7. \\ 100 | We know $\mathbb{E}[X^{T}A^{T}AX] = \sum_{i=1}^{k}\mathbb{E}[(A_{i}X)^{T}(A_iX)]$\\ 101 | On the other hand, 102 | \[ ||A\mu||_{2}^{2} + Tr(A\sum A^{T}) = \sum_{i=1}^{k}A_{i}^{T}\mu \mu^{T}A_{i} + Cov(A_{i}^{T}X, A_{i}^{T}X) = \sum_{i=1}^{k}A_{i}^{T}\mu \mu^{T}A_{i} + \mathbb{E}[A_{i}^{T}XX^{T}A_{i}] - A_{i}^{T}\mu \mu^{T} A_{i}\] 103 | \[ = \sum_{i=1}^{k}\mathbb{E}[A_{i}^{T}XX^{T}A_{i}] = \sum_{i=1}^{k}\mathbb{E}[(A_{i}X)^{T}(A_iX)]\] 104 | 105 | 106 | 107 | \end{problem} 108 | 109 | \end{document} -------------------------------------------------------------------------------- /PS11/ProblemSet11.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 4 | \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined 5 | \global\let\oldcontentsline\contentsline 6 | \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} 7 | \global\let\oldnewlabel\newlabel 8 | \gdef\newlabel#1#2{\newlabelxx{#1}#2} 9 | \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} 10 | \AtEndDocument{\ifx\hyper@anchor\@undefined 11 | \let\contentsline\oldcontentsline 12 | \let\newlabel\oldnewlabel 13 | \fi} 14 | \fi} 15 | \global\let\hyper@last\relax 16 | \gdef\HyperFirstAtBeginDocument#1{#1} 17 | \providecommand\HyField@AuxAddToFields[1]{} 18 | \providecommand\HyField@AuxAddToCoFields[2]{} 19 | -------------------------------------------------------------------------------- /PS11/ProblemSet11.log: -------------------------------------------------------------------------------- 1 | This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex 2020.2.15) 26 FEB 2020 21:18 2 | entering extended mode 3 | restricted \write18 enabled. 4 | %&-line parsing enabled. 5 | **ProblemSet11.tex 6 | (./ProblemSet11.tex 7 | LaTeX2e <2017-04-15> 8 | Babel <3.18> and hyphenation patterns for 5 language(s) loaded. 9 | (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls 10 | Document Class: article 2014/09/29 v1.4h Standard LaTeX document class 11 | (/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo 12 | File: size10.clo 2014/09/29 v1.4h Standard LaTeX file (size option) 13 | ) 14 | \c@part=\count79 15 | \c@section=\count80 16 | \c@subsection=\count81 17 | \c@subsubsection=\count82 18 | \c@paragraph=\count83 19 | \c@subparagraph=\count84 20 | \c@figure=\count85 21 | \c@table=\count86 22 | \abovecaptionskip=\skip41 23 | \belowcaptionskip=\skip42 24 | \bibindent=\dimen102 25 | ) 26 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty 27 | Package: hyperref 2018/02/06 v6.86b Hypertext links for LaTeX 28 | 29 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty 30 | Package: hobsub-hyperref 2016/05/16 v1.14 Bundle oberdiek, subset hyperref (HO) 31 | 32 | 33 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty 34 | Package: hobsub-generic 2016/05/16 v1.14 Bundle oberdiek, subset generic (HO) 35 | Package: hobsub 2016/05/16 v1.14 Construct package bundles (HO) 36 | Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) 37 | Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) 38 | Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) 39 | Package ifluatex Info: LuaTeX not detected. 40 | Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) 41 | Package ifvtex Info: VTeX not detected. 42 | Package: intcalc 2016/05/16 v1.2 Expandable calculations with integers (HO) 43 | Package: ifpdf 2017/03/15 v3.2 Provides the ifpdf switch 44 | Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) 45 | Package etexcmds Info: Could not find \expanded. 46 | (etexcmds) That can mean that you are not using pdfTeX 1.50 or 47 | (etexcmds) that some package has redefined \expanded. 48 | (etexcmds) In the latter case, load this package earlier. 49 | Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) 50 | Package: kvdefinekeys 2016/05/16 v1.4 Define keys (HO) 51 | Package: pdftexcmds 2018/01/21 v0.26 Utility functions of pdfTeX for LuaTeX (HO 52 | ) 53 | Package pdftexcmds Info: LuaTeX not detected. 54 | Package pdftexcmds Info: \pdf@primitive is available. 55 | Package pdftexcmds Info: \pdf@ifprimitive is available. 56 | Package pdftexcmds Info: \pdfdraftmode found. 57 | Package: pdfescape 2016/05/16 v1.14 Implements pdfTeX's escape features (HO) 58 | Package: bigintcalc 2016/05/16 v1.4 Expandable calculations on big integers (HO 59 | ) 60 | Package: bitset 2016/05/16 v1.2 Handle bit-vector datatype (HO) 61 | Package: uniquecounter 2016/05/16 v1.3 Provide unlimited unique counter (HO) 62 | ) 63 | Package hobsub Info: Skipping package `hobsub' (already loaded). 64 | Package: letltxmacro 2016/05/16 v1.5 Let assignment for LaTeX macros (HO) 65 | Package: hopatch 2016/05/16 v1.3 Wrapper for package hooks (HO) 66 | Package: xcolor-patch 2016/05/16 xcolor patch 67 | Package: atveryend 2016/05/16 v1.9 Hooks at the very end of document (HO) 68 | Package atveryend Info: \enddocument detected (standard20110627). 69 | Package: atbegshi 2016/06/09 v1.18 At begin shipout hook (HO) 70 | Package: refcount 2016/05/16 v3.5 Data extraction from label references (HO) 71 | Package: hycolor 2016/05/16 v1.8 Color options for hyperref/bookmark (HO) 72 | ) 73 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty 74 | Package: keyval 2014/10/28 v1.15 key=value parser (DPC) 75 | \KV@toks@=\toks14 76 | ) 77 | (/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty 78 | Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional 79 | ) 80 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty 81 | Package: auxhook 2016/05/16 v1.4 Hooks for auxiliary files (HO) 82 | ) 83 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty 84 | Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) 85 | ) 86 | \@linkdim=\dimen103 87 | \Hy@linkcounter=\count87 88 | \Hy@pagecounter=\count88 89 | 90 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def 91 | File: pd1enc.def 2018/02/06 v6.86b Hyperref: PDFDocEncoding definition (HO) 92 | ) 93 | \Hy@SavedSpaceFactor=\count89 94 | 95 | (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg 96 | File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive 97 | ) 98 | Package hyperref Info: Hyper figures OFF on input line 4509. 99 | Package hyperref Info: Link nesting OFF on input line 4514. 100 | Package hyperref Info: Hyper index ON on input line 4517. 101 | Package hyperref Info: Plain pages OFF on input line 4524. 102 | Package hyperref Info: Backreferencing OFF on input line 4529. 103 | Package hyperref Info: Implicit mode ON; LaTeX internals redefined. 104 | Package hyperref Info: Bookmarks ON on input line 4762. 105 | \c@Hy@tempcnt=\count90 106 | 107 | (/usr/share/texlive/texmf-dist/tex/latex/url/url.sty 108 | \Urlmuskip=\muskip10 109 | Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. 110 | ) 111 | LaTeX Info: Redefining \url on input line 5115. 112 | \XeTeXLinkMargin=\dimen104 113 | \Fld@menulength=\count91 114 | \Field@Width=\dimen105 115 | \Fld@charsize=\dimen106 116 | Package hyperref Info: Hyper figures OFF on input line 6369. 117 | Package hyperref Info: Link nesting OFF on input line 6374. 118 | Package hyperref Info: Hyper index ON on input line 6377. 119 | Package hyperref Info: backreferencing OFF on input line 6384. 120 | Package hyperref Info: Link coloring OFF on input line 6389. 121 | Package hyperref Info: Link coloring with OCG OFF on input line 6394. 122 | Package hyperref Info: PDF/A mode OFF on input line 6399. 123 | LaTeX Info: Redefining \ref on input line 6439. 124 | LaTeX Info: Redefining \pageref on input line 6443. 125 | \Hy@abspage=\count92 126 | \c@Item=\count93 127 | \c@Hfootnote=\count94 128 | ) 129 | Package hyperref Info: Driver (autodetected): hpdftex. 130 | 131 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def 132 | File: hpdftex.def 2018/02/06 v6.86b Hyperref driver for pdfTeX 133 | \Fld@listcount=\count95 134 | \c@bookmark@seq@number=\count96 135 | 136 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty 137 | Package: rerunfilecheck 2016/05/16 v1.8 Rerun checks for auxiliary files (HO) 138 | Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 139 | 82. 140 | ) 141 | \Hy@SectionHShift=\skip43 142 | ) 143 | (/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty 144 | Package: geometry 2010/09/12 v5.6 Page Geometry 145 | \Gm@cnth=\count97 146 | \Gm@cntv=\count98 147 | \c@Gm@tempcnt=\count99 148 | \Gm@bindingoffset=\dimen107 149 | \Gm@wd@mp=\dimen108 150 | \Gm@odd@mp=\dimen109 151 | \Gm@even@mp=\dimen110 152 | \Gm@layoutwidth=\dimen111 153 | \Gm@layoutheight=\dimen112 154 | \Gm@layouthoffset=\dimen113 155 | \Gm@layoutvoffset=\dimen114 156 | \Gm@dimlist=\toks15 157 | ) 158 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty 159 | Package: amsmath 2017/09/02 v2.17a AMS math features 160 | \@mathmargin=\skip44 161 | 162 | For additional information on amsmath, use the `?' option. 163 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty 164 | Package: amstext 2000/06/29 v2.01 AMS text 165 | 166 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty 167 | File: amsgen.sty 1999/11/30 v2.0 generic functions 168 | \@emptytoks=\toks16 169 | \ex@=\dimen115 170 | )) 171 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty 172 | Package: amsbsy 1999/11/29 v1.2d Bold Symbols 173 | \pmbraise@=\dimen116 174 | ) 175 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty 176 | Package: amsopn 2016/03/08 v2.02 operator names 177 | ) 178 | \inf@bad=\count100 179 | LaTeX Info: Redefining \frac on input line 213. 180 | \uproot@=\count101 181 | \leftroot@=\count102 182 | LaTeX Info: Redefining \overline on input line 375. 183 | \classnum@=\count103 184 | \DOTSCASE@=\count104 185 | LaTeX Info: Redefining \ldots on input line 472. 186 | LaTeX Info: Redefining \dots on input line 475. 187 | LaTeX Info: Redefining \cdots on input line 596. 188 | \Mathstrutbox@=\box26 189 | \strutbox@=\box27 190 | \big@size=\dimen117 191 | LaTeX Font Info: Redeclaring font encoding OML on input line 712. 192 | LaTeX Font Info: Redeclaring font encoding OMS on input line 713. 193 | \macc@depth=\count105 194 | \c@MaxMatrixCols=\count106 195 | \dotsspace@=\muskip11 196 | \c@parentequation=\count107 197 | \dspbrk@lvl=\count108 198 | \tag@help=\toks17 199 | \row@=\count109 200 | \column@=\count110 201 | \maxfields@=\count111 202 | \andhelp@=\toks18 203 | \eqnshift@=\dimen118 204 | \alignsep@=\dimen119 205 | \tagshift@=\dimen120 206 | \tagwidth@=\dimen121 207 | \totwidth@=\dimen122 208 | \lineht@=\dimen123 209 | \@envbody=\toks19 210 | \multlinegap=\skip45 211 | \multlinetaggap=\skip46 212 | \mathdisplay@stack=\toks20 213 | LaTeX Info: Redefining \[ on input line 2817. 214 | LaTeX Info: Redefining \] on input line 2818. 215 | ) 216 | (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty 217 | Package: amsthm 2017/10/31 v2.20.4 218 | \thm@style=\toks21 219 | \thm@bodyfont=\toks22 220 | \thm@headfont=\toks23 221 | \thm@notefont=\toks24 222 | \thm@headpunct=\toks25 223 | \thm@preskip=\skip47 224 | \thm@postskip=\skip48 225 | \thm@headsep=\skip49 226 | \dth@everypar=\toks26 227 | ) 228 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty 229 | Package: amssymb 2013/01/14 v3.01 AMS font symbols 230 | 231 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty 232 | Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support 233 | \symAMSa=\mathgroup4 234 | \symAMSb=\mathgroup5 235 | LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' 236 | (Font) U/euf/m/n --> U/euf/b/n on input line 106. 237 | )) 238 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty 239 | Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) 240 | 241 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty 242 | Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR) 243 | 244 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty 245 | Package: trig 2016/01/03 v1.10 sin cos tan (DPC) 246 | ) 247 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg 248 | File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration 249 | ) 250 | Package graphics Info: Driver file: pdftex.def on input line 99. 251 | 252 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def 253 | File: pdftex.def 2018/01/08 v1.0l Graphics/color driver for pdftex 254 | )) 255 | \Gin@req@height=\dimen124 256 | \Gin@req@width=\dimen125 257 | ) 258 | (/usr/share/texlive/texmf-dist/tex/latex/tools/multicol.sty 259 | Package: multicol 2017/04/11 v1.8q multicolumn formatting (FMi) 260 | \c@tracingmulticols=\count112 261 | \mult@box=\box28 262 | \multicol@leftmargin=\dimen126 263 | \c@unbalance=\count113 264 | \c@collectmore=\count114 265 | \doublecol@number=\count115 266 | \multicoltolerance=\count116 267 | \multicolpretolerance=\count117 268 | \full@width=\dimen127 269 | \page@free=\dimen128 270 | \premulticols=\dimen129 271 | \postmulticols=\dimen130 272 | \multicolsep=\skip50 273 | \multicolbaselineskip=\skip51 274 | \partial@page=\box29 275 | \last@line=\box30 276 | \maxbalancingoverflow=\dimen131 277 | \mult@rightbox=\box31 278 | \mult@grightbox=\box32 279 | \mult@gfirstbox=\box33 280 | \mult@firstbox=\box34 281 | \@tempa=\box35 282 | \@tempa=\box36 283 | \@tempa=\box37 284 | \@tempa=\box38 285 | \@tempa=\box39 286 | \@tempa=\box40 287 | \@tempa=\box41 288 | \@tempa=\box42 289 | \@tempa=\box43 290 | \@tempa=\box44 291 | \@tempa=\box45 292 | \@tempa=\box46 293 | \@tempa=\box47 294 | \@tempa=\box48 295 | \@tempa=\box49 296 | \@tempa=\box50 297 | \@tempa=\box51 298 | \c@columnbadness=\count118 299 | \c@finalcolumnbadness=\count119 300 | \last@try=\dimen132 301 | \multicolovershoot=\dimen133 302 | \multicolundershoot=\dimen134 303 | \mult@nat@firstbox=\box52 304 | \colbreak@box=\box53 305 | \mc@col@check@num=\count120 306 | ) 307 | (/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty 308 | Package: array 2016/10/06 v2.4d Tabular extension package (FMi) 309 | \col@sep=\dimen135 310 | \extrarowheight=\dimen136 311 | \NC@list=\toks27 312 | \extratabsurround=\skip52 313 | \backup@length=\skip53 314 | ) (./ProblemSet11.aux) 315 | \openout1 = `ProblemSet11.aux'. 316 | 317 | LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 14. 318 | LaTeX Font Info: ... okay on input line 14. 319 | LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 14. 320 | LaTeX Font Info: ... okay on input line 14. 321 | LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 14. 322 | LaTeX Font Info: ... okay on input line 14. 323 | LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 14. 324 | LaTeX Font Info: ... okay on input line 14. 325 | LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 14. 326 | LaTeX Font Info: ... okay on input line 14. 327 | LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 14. 328 | LaTeX Font Info: ... okay on input line 14. 329 | LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 14. 330 | LaTeX Font Info: ... okay on input line 14. 331 | \AtBeginShipoutBox=\box54 332 | Package hyperref Info: Link coloring OFF on input line 14. 333 | 334 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty 335 | Package: nameref 2016/05/21 v2.44 Cross-referencing by name of section 336 | 337 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty 338 | Package: gettitlestring 2016/05/16 v1.5 Cleanup title references (HO) 339 | ) 340 | \c@section@level=\count121 341 | ) 342 | LaTeX Info: Redefining \ref on input line 14. 343 | LaTeX Info: Redefining \pageref on input line 14. 344 | LaTeX Info: Redefining \nameref on input line 14. 345 | 346 | (./ProblemSet11.out) (./ProblemSet11.out) 347 | \@outlinefile=\write3 348 | \openout3 = `ProblemSet11.out'. 349 | 350 | 351 | *geometry* driver: auto-detecting 352 | *geometry* detected driver: pdftex 353 | *geometry* verbose mode - [ preamble ] result: 354 | * driver: pdftex 355 | * paper: 356 | * layout: 357 | * layoutoffset:(h,v)=(0.0pt,0.0pt) 358 | * modes: 359 | * h-part:(L,W,R)=(72.26999pt, 469.75502pt, 72.26999pt) 360 | * v-part:(T,H,B)=(72.26999pt, 650.43001pt, 72.26999pt) 361 | * \paperwidth=614.295pt 362 | * \paperheight=794.96999pt 363 | * \textwidth=469.75502pt 364 | * \textheight=650.43001pt 365 | * \oddsidemargin=0.0pt 366 | * \evensidemargin=0.0pt 367 | * \topmargin=-37.0pt 368 | * \headheight=12.0pt 369 | * \headsep=25.0pt 370 | * \topskip=10.0pt 371 | * \footskip=30.0pt 372 | * \marginparwidth=65.0pt 373 | * \marginparsep=11.0pt 374 | * \columnsep=10.0pt 375 | * \skip\footins=9.0pt plus 4.0pt minus 2.0pt 376 | * \hoffset=0.0pt 377 | * \voffset=0.0pt 378 | * \mag=1000 379 | * \@twocolumnfalse 380 | * \@twosidefalse 381 | * \@mparswitchfalse 382 | * \@reversemarginfalse 383 | * (1in=72.27pt=25.4mm, 1cm=28.453pt) 384 | 385 | (/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii 386 | [Loading MPS to PDF converter (version 2006.09.02).] 387 | \scratchcounter=\count122 388 | \scratchdimen=\dimen137 389 | \scratchbox=\box55 390 | \nofMPsegments=\count123 391 | \nofMParguments=\count124 392 | \everyMPshowfont=\toks28 393 | \MPscratchCnt=\count125 394 | \MPscratchDim=\dimen138 395 | \MPnumerator=\count126 396 | \makeMPintoPDFobject=\count127 397 | \everyMPtoPDFconversion=\toks29 398 | ) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty 399 | Package: epstopdf-base 2016/05/15 v2.6 Base part for package epstopdf 400 | 401 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/grfext.sty 402 | Package: grfext 2016/05/16 v1.2 Manage graphics extensions (HO) 403 | ) 404 | Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 405 | 38. 406 | Package grfext Info: Graphics extension search list: 407 | (grfext) [.pdf,.png,.jpg,.mps,.jpeg,.jbig2,.jb2,.PDF,.PNG,.JPG,.JPE 408 | G,.JBIG2,.JB2,.eps] 409 | (grfext) \AppendGraphicsExtensions on input line 456. 410 | 411 | (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg 412 | File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv 413 | e 414 | )) 415 | LaTeX Font Info: Try loading font information for U+msa on input line 18. 416 | 417 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd 418 | File: umsa.fd 2013/01/14 v3.01 AMS symbols A 419 | ) 420 | LaTeX Font Info: Try loading font information for U+msb on input line 18. 421 | 422 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd 423 | File: umsb.fd 2013/01/14 v3.01 AMS symbols B 424 | ) [1 425 | 426 | {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] 427 | Overfull \hbox (5.74615pt too wide) detected at line 63 428 | \U/msb/m/n/10 E\OT1/cmr/m/n/10 [[]\OML/cmm/m/it/10 log\OT1/cmr/m/n/10 (\OML/cmm 429 | /m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ))] = [] []\OML/ 430 | cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) = [] []\OML 431 | /cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) + [] [][]\ 432 | OML/cmm/m/it/10 dx[] \OT1/cmr/m/n/10 = \U/msb/m/n/10 E\OT1/cmr/m/n/10 [([])[]] 433 | [] 434 | 435 | Package atveryend Info: Empty hook `BeforeClearDocument' on input line 92. 436 | [2] 437 | Package atveryend Info: Empty hook `AfterLastShipout' on input line 92. 438 | (./ProblemSet11.aux) 439 | Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 92. 440 | Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 92. 441 | Package rerunfilecheck Info: File `ProblemSet11.out' has not changed. 442 | (rerunfilecheck) Checksum: D41D8CD98F00B204E9800998ECF8427E;0. 443 | Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 92. 444 | ) 445 | Here is how much of TeX's memory you used: 446 | 6545 strings out of 494880 447 | 93780 string characters out of 6179601 448 | 188401 words of memory out of 5000000 449 | 9807 multiletter control sequences out of 15000+600000 450 | 10591 words of font info for 42 fonts, out of 8000000 for 9000 451 | 36 hyphenation exceptions out of 8191 452 | 28i,10n,32p,515b,387s stack positions out of 5000i,500n,10000p,200000b,80000s 453 | < 464 | /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb> 467 | Output written on ProblemSet11.pdf (2 pages, 138579 bytes). 468 | PDF statistics: 469 | 77 PDF objects out of 1000 (max. 8388607) 470 | 57 compressed objects within 1 object stream 471 | 3 named destinations out of 1000 (max. 500000) 472 | 1 words of extra memory for PDF output out of 10000 (max. 10000000) 473 | 474 | -------------------------------------------------------------------------------- /PS11/ProblemSet11.out: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS11/ProblemSet11.out -------------------------------------------------------------------------------- /PS11/ProblemSet11.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS11/ProblemSet11.pdf -------------------------------------------------------------------------------- /PS11/ProblemSet11.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS11/ProblemSet11.synctex.gz -------------------------------------------------------------------------------- /PS11/ProblemSet11.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt]{article} 2 | \usepackage{hyperref} 3 | \usepackage[margin=1in]{geometry} 4 | \usepackage{amsmath,amsthm,amssymb, graphicx, multicol, array} 5 | 6 | \newcommand{\N}{\mathbb{N}} 7 | \newcommand{\Z}{\mathbb{Z}} 8 | 9 | \newenvironment{problem}[2][Problem]{\begin{trivlist} 10 | \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} 11 | 12 | 13 | 14 | \begin{document} 15 | \title{Problem Set 10} 16 | \author{Hoang Nguyen, Huy Nguyen} 17 | \maketitle 18 | 19 | \begin{problem}{1} 20 | \item 1. 21 | We have PDF: $p(x) = p^{x}(1-p)^{1-x} = e^{xlog(p) + (1-x)log(1-p)}$. Hence we have: 22 | \[\begin{cases} \eta_{1}=log(p) \\ \eta_{2} = log(1-p) \\ T_{1}(x)= x \\ T_{2}(x)=1-x \\ h(x) = 1 \\ B(p)=0 \end{cases} \] 23 | 24 | 25 | 26 | 27 | \item 2. 28 | We have PDF $p(x) = e^{\mu x- \frac{\mu^2}{2}}\frac{e^{\frac{-x^{2}}{2}}}{\sqrt{2\pi}}$.Hence 29 | \[\begin{cases} \eta=\mu \\ T(x)= x \\ h(x) = \frac{e^{\frac{-x^{2}}{2}}}{\sqrt{2\pi}} \\ B(p)=\frac{\mu^2}{2} \end{cases} \] 30 | 31 | 32 | 33 | 34 | \item 3. 35 | We have PDF $p(x) = e^{\frac{\mu}{\sigma^2} - \frac{1}{2\sigma^2}x^{2} - \frac{\mu^2}{2\sigma^2}}\frac{1}{\sigma\sqrt{2\pi}}$. Hence 36 | \[ \begin{cases} \eta_{1}=\frac{\mu}{\sigma^2} \\ \eta_{2} = -\frac{1}{2\sigma^2} \\ T_{1}(x)= x \\ T_{2}(x)=x^{2} \\ h(x) = 1 \\ B(p)=\frac{\mu^2}{2\sigma^2} + log(\sigma\sqrt{2\pi}) \end{cases} \] 37 | 38 | \item 4. 39 | We have $p(x) = \lambda e^{-\lambda x}$. Hence 40 | \[ \begin{cases} \eta=-\lambda \\ T(x)= x \\ h(x) = \lambda \\ B(p)=0 \end{cases}\] 41 | 42 | \item 5. 43 | We have $p(x)= \frac{1}{\upsilon} = e^{-log(\upsilon)}$. Hence 44 | We have $p(x) = \lambda e^{-\lambda x}$. Hence 45 | \[ \begin{cases} \eta=-log(\upsilon) \\ T(x)= 1 \\ h(x) = 1 \\ B(p)=0 \end{cases}\] 46 | 47 | \item 6. 48 | We have $p(x) = \frac{\beta^{\alpha}}{\mathbb{I}(\alpha)}x^{\alpha - 1}e^{-\beta x} = e^{-\beta x + (\alpha -1)log(x)}\frac{\beta^{\alpha}}{\mathbb{I}(\alpha)}$. Hence 49 | \[\begin{cases} \eta_{1}=-\beta \\ \eta_{2} = \alpha -1 \\ T_{1}(x)= x \\ T_{2}(x)=log(x) \\ h(x) = 1 \\ B(p)= -log(\frac{\beta^{\alpha}}{\mathbb{I}(\alpha)}) \end{cases} \] 50 | 51 | \item 7. 52 | We have $p(x) = \frac{\lambda^{x} e^{-\lambda}}{x!} = e^{-\lambda + log(\lambda) x}\frac{1}{x!}$. Hence 53 | \[\begin{cases} \eta_{1}=log(\lambda)\\ T(x)= x \\ h(x) = \frac{1}{x!} \\ B(p)= -\lambda \end{cases} \] 54 | 55 | \end{problem} 56 | 57 | \begin{problem}{2} 58 | \item 1. \\ 59 | a) \\ 60 | b) We have 61 | \[\mathbb{E}[\frac{\partial log(f_{\theta}(x))}{\partial \theta}] = \mathbb{E}[\frac{\frac{\partial }{\partial \theta} f_{\theta}(x)}{f_{\theta}(x)}] = \int \frac{\frac{\partial }{\partial \theta} f_{\theta}(x)}{f_{\theta}(x)} f_{\theta}(x) = \frac{\partial }{\partial \theta} \int f_{\theta}(x)dx = 0 \] 62 | 63 | \[\mathbb{E}[\frac{\partial^2}{\partial \theta^2} log(f_{\theta}(x))] = \int \frac{[\frac{\partial^2}{\partial \theta^2} f_{\theta}(x)] f_{\theta}(x) - [\frac{\partial}{\partial \theta} f_{\theta}(x)]^2 }{(f_{\theta}(x))^2} f_{\theta}(x) = \int \frac{\partial^2}{\partial \theta^2} f_{\theta}(x) + \int \Big ( \frac{[\frac{\partial}{\partial \theta} f_{\theta}(x)]^2}{f_{\theta}(x)}dx \Big ) = \mathbb{E}[(\frac{\frac{\partial}{\partial \theta}f_{\theta} (x)}{f_{\theta (x)}})^2 ] \] 64 | We have $l(\theta) = -\theta x + b(\theta) + log(a(x))$. Hence $l'(\theta) = -x + b'(\theta) $ and $l''(\theta) = b''(\theta)$. From previous quantities, $\mathbb{E}[l'(\theta)]= - \mathbb{E}[x] + b'(\theta) =0 $\\ 65 | From previous quantities, $\mathbb{E}[\frac{\partial^2}{\partial \theta^2} log(f_{\theta}(x))] = \mathbb{E}[l''(\theta)] = \mathbb{E}[(l'(\theta))^2] $. Because $\mathbb{E}[l'(\theta)] = 0$, hence $\mathbb{E}[(l'(\theta))^2] = var(l'(\theta)) = - \mathbb{E}[l''(\theta)]$\\ 66 | c) Fisher Information. 67 | \item 2.\\ 68 | From previous questions, we have $l'(\theta) = -x + b'(\theta)$ and $l''(\theta) = b''(\theta)$ 69 | \item 3. \\ 70 | From previous result $ \mathbb{E}[l'(\theta)]-\mathbb{E}[x] + b'(\theta)=0 \Rightarrow \mathbb{E}[x]=b'(\theta)$\\ 71 | From previous result $\mathbb{E}[(l'(\theta))^2] = \mathbb{E}[(-x + b'(\theta))^2] = \mathbb{E}[(-x + E[x])^2] = var(x) =- b''(\theta)$ 72 | \item 4. \\ 73 | a) Sample space $X=(0, + \infty)$\\ 74 | b) $a(x)= x^{\alpha -1}$ and $b(\theta) = log(\frac{\theta^{\alpha}}{\mathbb{I}(\alpha)})$ 75 | c) From previous result, $\mathbb{E}[x]=b'(\theta)= \frac{\alpha}{\theta}$ and $Var(x)=-b''(\theta)= \frac{\alpha}{\theta^2}$ 76 | 77 | \end{problem} 78 | \begin{problem}{3} 79 | \item 1. $Z_{1}$ is either 0 or 1, hence $Z_{1}$ is Bernoulli distribution conditional on $X_{i}$ with parameter $p=1-F(-X^{T}\beta)$. 80 | \item 2. We have $\mathbb{E}[Z]=p=F(-X^{T}\beta)$. Hence $g(1-F(-X^{T}\beta)) = X^{T}\beta \Rightarrow g(x)= (1-F(-x))^{-1}$. 81 | \item 3. If $\epsilon$ is standard Gaussian, then $g(x)=(1-F(-x))^{-1}=(1-\phi(-x))^{-1}=\phi(x)^{-1}$. 82 | \item 4. \\ 83 | a) $F(t)= \int \frac{e^{-t}}{(1+e^{-1})^2} = -\frac{e^{-t}}{1+e^{-t}}$\\ 84 | b) $g(x)= 1-F^{-1}(x)=1-log(\frac{x}{1-x})$\\ 85 | c) 86 | 87 | 88 | 89 | 90 | \end{problem} 91 | 92 | \end{document} -------------------------------------------------------------------------------- /PS2/Problem_set2.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | -------------------------------------------------------------------------------- /PS2/Problem_set2.log: -------------------------------------------------------------------------------- 1 | This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex 2019.9.8) 16 SEP 2019 17:09 2 | entering extended mode 3 | restricted \write18 enabled. 4 | %&-line parsing enabled. 5 | **Problem_set2.tex 6 | (./Problem_set2.tex 7 | LaTeX2e <2017-04-15> 8 | Babel <3.18> and hyphenation patterns for 84 language(s) loaded. 9 | (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls 10 | Document Class: article 2014/09/29 v1.4h Standard LaTeX document class 11 | (/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo 12 | File: size10.clo 2014/09/29 v1.4h Standard LaTeX file (size option) 13 | ) 14 | \c@part=\count79 15 | \c@section=\count80 16 | \c@subsection=\count81 17 | \c@subsubsection=\count82 18 | \c@paragraph=\count83 19 | \c@subparagraph=\count84 20 | \c@figure=\count85 21 | \c@table=\count86 22 | \abovecaptionskip=\skip41 23 | \belowcaptionskip=\skip42 24 | \bibindent=\dimen102 25 | ) 26 | (/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty 27 | Package: geometry 2010/09/12 v5.6 Page Geometry 28 | 29 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty 30 | Package: keyval 2014/10/28 v1.15 key=value parser (DPC) 31 | \KV@toks@=\toks14 32 | ) 33 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty 34 | Package: ifpdf 2017/03/15 v3.2 Provides the ifpdf switch 35 | ) 36 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifvtex.sty 37 | Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) 38 | Package ifvtex Info: VTeX not detected. 39 | ) 40 | (/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty 41 | Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional 42 | ) 43 | \Gm@cnth=\count87 44 | \Gm@cntv=\count88 45 | \c@Gm@tempcnt=\count89 46 | \Gm@bindingoffset=\dimen103 47 | \Gm@wd@mp=\dimen104 48 | \Gm@odd@mp=\dimen105 49 | \Gm@even@mp=\dimen106 50 | \Gm@layoutwidth=\dimen107 51 | \Gm@layoutheight=\dimen108 52 | \Gm@layouthoffset=\dimen109 53 | \Gm@layoutvoffset=\dimen110 54 | \Gm@dimlist=\toks15 55 | ) 56 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty 57 | Package: amsmath 2017/09/02 v2.17a AMS math features 58 | \@mathmargin=\skip43 59 | 60 | For additional information on amsmath, use the `?' option. 61 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty 62 | Package: amstext 2000/06/29 v2.01 AMS text 63 | 64 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty 65 | File: amsgen.sty 1999/11/30 v2.0 generic functions 66 | \@emptytoks=\toks16 67 | \ex@=\dimen111 68 | )) 69 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty 70 | Package: amsbsy 1999/11/29 v1.2d Bold Symbols 71 | \pmbraise@=\dimen112 72 | ) 73 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty 74 | Package: amsopn 2016/03/08 v2.02 operator names 75 | ) 76 | \inf@bad=\count90 77 | LaTeX Info: Redefining \frac on input line 213. 78 | \uproot@=\count91 79 | \leftroot@=\count92 80 | LaTeX Info: Redefining \overline on input line 375. 81 | \classnum@=\count93 82 | \DOTSCASE@=\count94 83 | LaTeX Info: Redefining \ldots on input line 472. 84 | LaTeX Info: Redefining \dots on input line 475. 85 | LaTeX Info: Redefining \cdots on input line 596. 86 | \Mathstrutbox@=\box26 87 | \strutbox@=\box27 88 | \big@size=\dimen113 89 | LaTeX Font Info: Redeclaring font encoding OML on input line 712. 90 | LaTeX Font Info: Redeclaring font encoding OMS on input line 713. 91 | \macc@depth=\count95 92 | \c@MaxMatrixCols=\count96 93 | \dotsspace@=\muskip10 94 | \c@parentequation=\count97 95 | \dspbrk@lvl=\count98 96 | \tag@help=\toks17 97 | \row@=\count99 98 | \column@=\count100 99 | \maxfields@=\count101 100 | \andhelp@=\toks18 101 | \eqnshift@=\dimen114 102 | \alignsep@=\dimen115 103 | \tagshift@=\dimen116 104 | \tagwidth@=\dimen117 105 | \totwidth@=\dimen118 106 | \lineht@=\dimen119 107 | \@envbody=\toks19 108 | \multlinegap=\skip44 109 | \multlinetaggap=\skip45 110 | \mathdisplay@stack=\toks20 111 | LaTeX Info: Redefining \[ on input line 2817. 112 | LaTeX Info: Redefining \] on input line 2818. 113 | ) 114 | (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty 115 | Package: amsthm 2017/10/31 v2.20.4 116 | \thm@style=\toks21 117 | \thm@bodyfont=\toks22 118 | \thm@headfont=\toks23 119 | \thm@notefont=\toks24 120 | \thm@headpunct=\toks25 121 | \thm@preskip=\skip46 122 | \thm@postskip=\skip47 123 | \thm@headsep=\skip48 124 | \dth@everypar=\toks26 125 | ) 126 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty 127 | Package: amssymb 2013/01/14 v3.01 AMS font symbols 128 | 129 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty 130 | Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support 131 | \symAMSa=\mathgroup4 132 | \symAMSb=\mathgroup5 133 | LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' 134 | (Font) U/euf/m/n --> U/euf/b/n on input line 106. 135 | )) 136 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty 137 | Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) 138 | 139 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty 140 | Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR) 141 | 142 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty 143 | Package: trig 2016/01/03 v1.10 sin cos tan (DPC) 144 | ) 145 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg 146 | File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration 147 | ) 148 | Package graphics Info: Driver file: pdftex.def on input line 99. 149 | 150 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def 151 | File: pdftex.def 2018/01/08 v1.0l Graphics/color driver for pdftex 152 | )) 153 | \Gin@req@height=\dimen120 154 | \Gin@req@width=\dimen121 155 | ) 156 | (/usr/share/texlive/texmf-dist/tex/latex/tools/multicol.sty 157 | Package: multicol 2017/04/11 v1.8q multicolumn formatting (FMi) 158 | \c@tracingmulticols=\count102 159 | \mult@box=\box28 160 | \multicol@leftmargin=\dimen122 161 | \c@unbalance=\count103 162 | \c@collectmore=\count104 163 | \doublecol@number=\count105 164 | \multicoltolerance=\count106 165 | \multicolpretolerance=\count107 166 | \full@width=\dimen123 167 | \page@free=\dimen124 168 | \premulticols=\dimen125 169 | \postmulticols=\dimen126 170 | \multicolsep=\skip49 171 | \multicolbaselineskip=\skip50 172 | \partial@page=\box29 173 | \last@line=\box30 174 | \maxbalancingoverflow=\dimen127 175 | \mult@rightbox=\box31 176 | \mult@grightbox=\box32 177 | \mult@gfirstbox=\box33 178 | \mult@firstbox=\box34 179 | \@tempa=\box35 180 | \@tempa=\box36 181 | \@tempa=\box37 182 | \@tempa=\box38 183 | \@tempa=\box39 184 | \@tempa=\box40 185 | \@tempa=\box41 186 | \@tempa=\box42 187 | \@tempa=\box43 188 | \@tempa=\box44 189 | \@tempa=\box45 190 | \@tempa=\box46 191 | \@tempa=\box47 192 | \@tempa=\box48 193 | \@tempa=\box49 194 | \@tempa=\box50 195 | \@tempa=\box51 196 | \c@columnbadness=\count108 197 | \c@finalcolumnbadness=\count109 198 | \last@try=\dimen128 199 | \multicolovershoot=\dimen129 200 | \multicolundershoot=\dimen130 201 | \mult@nat@firstbox=\box52 202 | \colbreak@box=\box53 203 | \mc@col@check@num=\count110 204 | ) 205 | (/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty 206 | Package: array 2016/10/06 v2.4d Tabular extension package (FMi) 207 | \col@sep=\dimen131 208 | \extrarowheight=\dimen132 209 | \NC@list=\toks27 210 | \extratabsurround=\skip51 211 | \backup@length=\skip52 212 | ) (./Problem_set2.aux) 213 | \openout1 = `Problem_set2.aux'. 214 | 215 | LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 14. 216 | LaTeX Font Info: ... okay on input line 14. 217 | LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 14. 218 | LaTeX Font Info: ... okay on input line 14. 219 | LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 14. 220 | LaTeX Font Info: ... okay on input line 14. 221 | LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 14. 222 | LaTeX Font Info: ... okay on input line 14. 223 | LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 14. 224 | LaTeX Font Info: ... okay on input line 14. 225 | LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 14. 226 | LaTeX Font Info: ... okay on input line 14. 227 | 228 | *geometry* driver: auto-detecting 229 | *geometry* detected driver: pdftex 230 | *geometry* verbose mode - [ preamble ] result: 231 | * driver: pdftex 232 | * paper: 233 | * layout: 234 | * layoutoffset:(h,v)=(0.0pt,0.0pt) 235 | * modes: 236 | * h-part:(L,W,R)=(72.26999pt, 469.75502pt, 72.26999pt) 237 | * v-part:(T,H,B)=(72.26999pt, 650.43001pt, 72.26999pt) 238 | * \paperwidth=614.295pt 239 | * \paperheight=794.96999pt 240 | * \textwidth=469.75502pt 241 | * \textheight=650.43001pt 242 | * \oddsidemargin=0.0pt 243 | * \evensidemargin=0.0pt 244 | * \topmargin=-37.0pt 245 | * \headheight=12.0pt 246 | * \headsep=25.0pt 247 | * \topskip=10.0pt 248 | * \footskip=30.0pt 249 | * \marginparwidth=65.0pt 250 | * \marginparsep=11.0pt 251 | * \columnsep=10.0pt 252 | * \skip\footins=9.0pt plus 4.0pt minus 2.0pt 253 | * \hoffset=0.0pt 254 | * \voffset=0.0pt 255 | * \mag=1000 256 | * \@twocolumnfalse 257 | * \@twosidefalse 258 | * \@mparswitchfalse 259 | * \@reversemarginfalse 260 | * (1in=72.27pt=25.4mm, 1cm=28.453pt) 261 | 262 | (/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii 263 | [Loading MPS to PDF converter (version 2006.09.02).] 264 | \scratchcounter=\count111 265 | \scratchdimen=\dimen133 266 | \scratchbox=\box54 267 | \nofMPsegments=\count112 268 | \nofMParguments=\count113 269 | \everyMPshowfont=\toks28 270 | \MPscratchCnt=\count114 271 | \MPscratchDim=\dimen134 272 | \MPnumerator=\count115 273 | \makeMPintoPDFobject=\count116 274 | \everyMPtoPDFconversion=\toks29 275 | ) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty 276 | Package: epstopdf-base 2016/05/15 v2.6 Base part for package epstopdf 277 | 278 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty 279 | Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) 280 | ) 281 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/grfext.sty 282 | Package: grfext 2016/05/16 v1.2 Manage graphics extensions (HO) 283 | 284 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvdefinekeys.sty 285 | Package: kvdefinekeys 2016/05/16 v1.4 Define keys (HO) 286 | 287 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty 288 | Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) 289 | ))) 290 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty 291 | Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) 292 | 293 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty 294 | Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) 295 | 296 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/etexcmds.sty 297 | Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) 298 | 299 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifluatex.sty 300 | Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) 301 | Package ifluatex Info: LuaTeX not detected. 302 | ) 303 | Package etexcmds Info: Could not find \expanded. 304 | (etexcmds) That can mean that you are not using pdfTeX 1.50 or 305 | (etexcmds) that some package has redefined \expanded. 306 | (etexcmds) In the latter case, load this package earlier. 307 | ))) 308 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/pdftexcmds.sty 309 | Package: pdftexcmds 2018/01/21 v0.26 Utility functions of pdfTeX for LuaTeX (HO 310 | ) 311 | Package pdftexcmds Info: LuaTeX not detected. 312 | Package pdftexcmds Info: \pdf@primitive is available. 313 | Package pdftexcmds Info: \pdf@ifprimitive is available. 314 | Package pdftexcmds Info: \pdfdraftmode found. 315 | ) 316 | Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 317 | 38. 318 | Package grfext Info: Graphics extension search list: 319 | (grfext) [.pdf,.png,.jpg,.mps,.jpeg,.jbig2,.jb2,.PDF,.PNG,.JPG,.JPE 320 | G,.JBIG2,.JB2,.eps] 321 | (grfext) \AppendGraphicsExtensions on input line 456. 322 | 323 | (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg 324 | File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv 325 | e 326 | )) 327 | LaTeX Font Info: Try loading font information for U+msa on input line 17. 328 | 329 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd 330 | File: umsa.fd 2013/01/14 v3.01 AMS symbols A 331 | ) 332 | LaTeX Font Info: Try loading font information for U+msb on input line 17. 333 | 334 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd 335 | File: umsb.fd 2013/01/14 v3.01 AMS symbols B 336 | ) 337 | Underfull \hbox (badness 10000) in paragraph at lines 22--23 338 | 339 | [] 340 | 341 | [1 342 | 343 | {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] 344 | (./Problem_set2.aux) ) 345 | Here is how much of TeX's memory you used: 346 | 3050 strings out of 492982 347 | 40909 string characters out of 6134897 348 | 116875 words of memory out of 5000000 349 | 6566 multiletter control sequences out of 15000+600000 350 | 11556 words of font info for 45 fonts, out of 8000000 for 9000 351 | 1141 hyphenation exceptions out of 8191 352 | 41i,16n,32p,279b,237s stack positions out of 5000i,500n,10000p,200000b,80000s 353 | 370 | 371 | Output written on Problem_set2.pdf (4 pages, 177040 bytes). 372 | PDF statistics: 373 | 89 PDF objects out of 1000 (max. 8388607) 374 | 64 compressed objects within 1 object stream 375 | 0 named destinations out of 1000 (max. 500000) 376 | 1 words of extra memory for PDF output out of 10000 (max. 10000000) 377 | 378 | -------------------------------------------------------------------------------- /PS2/Problem_set2.out: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS2/Problem_set2.out -------------------------------------------------------------------------------- /PS2/Problem_set2.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS2/Problem_set2.pdf -------------------------------------------------------------------------------- /PS2/Problem_set2.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS2/Problem_set2.synctex.gz -------------------------------------------------------------------------------- /PS2/Problem_set2.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt]{article} 2 | 3 | \usepackage[margin=1in]{geometry} 4 | \usepackage{amsmath,amsthm,amssymb, graphicx, multicol, array} 5 | 6 | \newcommand{\N}{\mathbb{N}} 7 | \newcommand{\Z}{\mathbb{Z}} 8 | 9 | \newenvironment{problem}[2][Problem]{\begin{trivlist} 10 | \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} 11 | 12 | 13 | 14 | \begin{document} 15 | \title{Problem Set 2} 16 | \author{Hoang Nguyen, Huy Nguyen} 17 | \maketitle 18 | 19 | \begin{problem}{1} 20 | \item 1. 21 | We know $\mathbb{E}[X_i]=1.p+ 0.(1-p)=p$ and $\mathbb{E}[X_i^2]=1.p+ 0.(1-p)=p$. Hence, $\mathbb{V}[X_i]= \mathbb{E}[X_i^2]- (\mathbb{E}[X_i])^2 =p(1-p).$ 22 | \item 2. \textit{Theorem}: If bias $\rightarrow$ 0 ans se (standard error) $\rightarrow$ 0 as n $\rightarrow \infty$ then $\hat{\theta}$ is consistent, that is, $\hat{\theta} \xrightarrow{\textbf{P}} \theta$.\\ 23 | 24 | 25 | 26 | 27 | $\mathbb{V}[\bar{X_i}(1-\bar{X_i})]=\mathbb{V}[\frac{\sum_{i=1}^{n} X_i}{n}(1-\frac{\sum_{i=1}^{n} X_i}{n})]=\mathbb{V}[\frac{1}{n^2} \sum_{i=1}^{n} X_i \Big(n-\sum_{i=1}^{n} X_i\Big) ]$ 28 | \[=\frac{1}{n^4} \mathbb{V}[\sum_{i=1}^{n} X_i \Big(n-\sum_{i=1}^{n} X_i\Big)]\] 29 | \[=\frac{1}{n^4} \mathbb{V}\Big[ \sum_{i=1}^{n} X_i \sum_{i=1}^{n} (1-X_i) \Big]\] 30 | 31 | \[=\frac{1}{n^4} \mathbb{V}\Big[\sum_{i=1}^{n} X_i(1-X_i) +\sum_{i \neq j} X_i(1-X_j) \Big] \] 32 | 33 | $X_i \in {0,1 } \Rightarrow \sum_{i=1}^{n} X_i(1-X_i)=0$ 34 | \\ 35 | Hence, 36 | 37 | \[\mathbb{V}[\bar{X_i}(1-\bar{X_i})]= \frac{1}{n^4} \mathbb{V} \Big[ \sum_{i \neq j} X_i(1-X_j) \Big] \] 38 | 39 | By $X_i$ is IID: 40 | \[\mathbb{V}[\bar{X_i}(1-\hat{\bar{X_i}})]= \frac{1}{n^4} \sum_{i \neq j} \mathbb{V} [X_i]\mathbb{V}[1-X_i] \] 41 | \[= \frac{1}{n^4} n(n-1)p(p-1) \] 42 | The last equality tends to 0 as n $\rightarrow \infty$. So we have $\mathbb{V}[\bar{X_i}(1-\bar{X_i})] \rightarrow$ 0. \\ 43 | Hence, se($\bar{X_i}(1-\bar{X_i})$)=$\sqrt{\mathbb{V}[\bar{X_i}(1-\bar{X_i})]} \rightarrow$ 0. (1)\\ 44 | By similar method, we can esaily get: $\mathbb{E}[\bar{X_i}(1-\bar{X_i})]= \frac{1}{n^2} \sum_{i \neq j} \mathbb{E}[X_i(1-X_j)] = \frac{n(n-1)}{n^2} p(1-p) \rightarrow p(1-p)$ as n $\rightarrow \infty$.\\ 45 | Hence, bias$\Big(\bar{X_i}(1-\bar{X_i})\Big)= \mathbb{E}[\bar{X_i}(1-\bar{X_i})] - p(1-p) \rightarrow 0$ as n $\rightarrow \infty$ (2)\\ 46 | From (1), (2) and the theorem above, we get $\bar{X_i}(1-\bar{X_i})$ is a consistent estimator of p(1-p). 47 | 48 | \item 3. We actually complete this exercise in previous solution.\\ 49 | bias$\Big(\bar{X_i}(1-\bar{X_i})\Big)= \mathbb{E}[\bar{X_i}(1-\bar{X_i})] - p(1-p)= \frac{n(n-1)}{n^2} p(1-p) -p(1-p)$. 50 | \item 4. To find an unbiased estimator, we have to find x such that $\frac{xn(n-1)}{n^2}=1 \Rightarrow x=\frac{n}{n-1}$.\\ 51 | Hence, an unbiased estimator can be $\frac{n}{n-1}\bar{X_i}(1-\bar{X_i})$ 52 | 53 | 54 | 55 | 56 | 57 | \end{problem} 58 | 59 | \begin{problem}{2} 60 | \item 1. 61 | $(\mathbb{N}, (Poiss(\lambda))_{\lambda>0})$. This paramter is identified. 62 | \item 2. 63 | $(\mathbb{R_+}, (Exp(\lambda))_{10>\lambda>0})$. This parameter is identified. 64 | \item 3. 65 | $(\mathbb{R_+}, (Uni(0, \theta))_{\theta >0})$. This parameter is identified. 66 | \item 4. 67 | $(\mathbb{R}, (N(\mu, \sigma^2))_{(\mu, \sigma^2) \in \mathbb{R} \times \mathbb{R_+}})$. These parater are identified. 68 | \item 5. 69 | \[\mathbb{P}(N(\mu, \sigma^2)>0)=\mathbb{P}\Big( N(0,1) > \frac{-\mu}{\sigma^2} \Big)=\phi(\frac{\mu}{\sigma^2})\] 70 | Hence, the statistical model is: $(\{0,1\}, (Ber(\phi(\frac{\mu}{\sigma^2}))_{(\mu, \sigma^2) \in \mathbb{R} \times \mathbb{R_+}})$. This model depends on $\frac{\mu}{\sigma^2} \Rightarrow$ these parameters are not identified. 71 | \item 6. 72 | Same for 3. 73 | \item 7. 74 | Let X $\thicksim Exp(\lambda) \Rightarrow \mathbb{P}(X>20)=e^{-20\lambda}$. Hence, the statistical model is: 75 | \[(\{ 0,1\},(Ber(e^{-20\lambda}))_{\lambda>0}) \] 76 | This parameter is identified. 77 | 78 | \item 8. 79 | Let X $\thicksim Ber(p)$ such that: 80 | \begin{align} 81 | \begin{cases} 82 | X_{i}=1 \text{ if machine i has timelife less than 500 days} \\ 83 | X_{i}=0 \text{ otherwise} 84 | \end{cases} 85 | \end{align} 86 | 87 | Hence: 88 | \[p=\mathbb{P}(X_{i}=1)=1-e^{-500\lambda}\] 89 | The number of machines that have stopped working before 500 days is a binominal random variable with parameter (67, $1-e^{-500\lambda}$)\\ 90 | The statistical model is $(\{1,2,3,..,67 \}, (Binominal(67, 1-e^{-500\lambda}))_{\lambda>0})$. This parameter is identified. 91 | 92 | \end{problem} 93 | 94 | \begin{problem}{3} 95 | \item 1. 96 | By central limit theorem (CLT), we have: 97 | \[ \frac{\sqrt{n}(\bar{X_i}-\mu)}{\sigma} \thicksim \text( N(0,1))\] 98 | Hence, $(a_{n})_{n \in \mathbb{N}}$ can be $\frac{\sqrt{n}}{\sigma}$ and $(b_{n})_{n \in \mathbb{N}}$ can be $\mu$. 99 | 100 | \item 2. 101 | We have: $Z \backsim N(0,1)$\\ 102 | Hence, $\mathbb{P}[|Z| \leqslant t]=\mathbb{P}[-t\leqslant Z \leqslant t]= \phi(t)-\phi(-t)= \phi(t)-(1-\phi(t))=2\phi(t)-1= 2\mathbb{P}[Z \leqslant t]-1$. 103 | \item 3. 104 | From part 1 we get: 105 | \[ \frac{\sqrt{n}(\bar{X_i}-\mu)}{\sigma} \thicksim \text( N(0,1))\] 106 | from part 2 we get: 107 | \[ \mathbb{P}[|Z| \leqslant t]=2\mathbb{P}[Z \leqslant t]-1\] 108 | Substitution: 109 | \[ \mathbb{P}\Big[|\frac{\sqrt{n}(\bar{X_i}-\mu)}{\sigma}| \leqslant t\Big]=2\mathbb{P}[Z \leqslant t]-1\] 110 | We have $2\mathbb{P}[Z \leqslant t]-1=0.95 \Rightarrow t=\phi^{-1}(\frac{0.95+1}{2})=1.96$. \\ 111 | Hence, 112 | \[ \mathbb{P}\Big[|\frac{\sqrt{n}(\bar{X_i}-\mu)}{\sigma}| \leqslant 1.96\Big]= 0.95\] 113 | 114 | \[ \mathbb{P}\Big[-\frac{\sqrt{n}(\bar{X_i}-\mu)}{\sigma} \leqslant 1.96 \leqslant \frac{\sqrt{n}(\bar{X_i}-\mu)}{\sigma}\Big]= 0.95\] 115 | Because $X_i$ is Poisson random variable with parameter $\lambda$, so $ \mu=\lambda$ and $\sigma= \sqrt{\lambda}$\\ 116 | We get: 117 | \[ \mathbb{P}\Big[-\frac{\sqrt{n}(\bar{X_i}-\lambda)}{\sqrt{\lambda}} \leqslant 1.96 \leqslant \frac{\sqrt{n}(\bar{X_i}-\lambda)}{\sqrt{\lambda}}\Big]= 0.95\] 118 | 119 | \[ \mathbb{P}\Big[\bar{X_i}- \frac{1.96\sqrt{\lambda}}{\sqrt{n}} \leqslant \lambda \leqslant \bar{X_i}+ \frac{1.96\sqrt{\lambda}}{\sqrt{n}} \Big] =0.95\] 120 | 121 | We know: $\bar{X_i} \xrightarrow{P} \mathbb{E}[\bar{X_i}]=\lambda$\\ 122 | Hence, 123 | \[ \mathbb{P}\Big[\bar{X_i}- \frac{1.96\sqrt{\bar{X_i}}}{\sqrt{n}} \leqslant \lambda \leqslant \bar{X_i}+ \frac{1.96\sqrt{\bar{X_i}}}{\sqrt{n}} \Big] \geqslant 0.95\] 124 | $\Rightarrow$ L=[$\bar{X_i}- \frac{1.96\sqrt{\bar{X_i}}}{\sqrt{n}}, \bar{X_i}+ \frac{1.96\sqrt{\bar{X_i}}}{\sqrt{n}}$] 125 | \item 4. 126 | We can easily see min($X_i$)$\leqslant \bar{X_i} \leqslant$ max($X_i$). Hence, a new interval can be: 127 | \[L=[min(X_i)- \frac{1.96\sqrt{\bar{X_i}}}{\sqrt{n}}, max(X_i)+ \frac{1.96\sqrt{\bar{X_i}}}{\sqrt{n}}]\] 128 | \end{problem} 129 | \begin{problem}{4} 130 | We have $X_i$ is IID. Hence, $\mathbb{P}(M_n \leqslant t)=\prod_{n=1}^{n}\mathbb{P}(X_i\leqslant t).$\\ 131 | By uniform distribution, the CDF of $M_n$: 132 | \[ \mathbb{P}(M_n \leqslant t)=F(t)=\Big(\frac{t}{\theta}\Big)^n \] 133 | Hence, the PDF of $M_n$ is: 134 | \[f(t)=\frac{dF}{dt}=n\theta^{-n}t^{n-1} \] 135 | We can easily get: 136 | \[\mathbb{E}[M_n]= \int_{0}^{\theta} tn\theta^{-n}t^{n-1}dt=\frac{n}{n+1}\theta \rightarrow \theta \textbf{ as n}\rightarrow \infty \] 137 | By Markov's Inequality: 138 | \[ \mathbb{P}\Big[ | M_n -\theta|> \epsilon \Big]\leqslant \mathbb{P}[M_n-\theta>\epsilon] \leqslant \frac{\mathbb{E}[M_n-\theta]}{\epsilon}=\frac{\mathbb{E}[M_n]-\theta}{\epsilon} \rightarrow 0\] 139 | Hence, $M_n$ converages in probility to $\theta$. 140 | 141 | \item 2. 142 | From part 1 we get: $M_n$: $\mathbb{P}[M_n \leqslant t]=\Big(\frac{t}{\theta}\Big)^n$. Hence, CDF of $n(1-\frac{M_n}{\theta})$ is: 143 | \[ P\Big[n(1-\frac{M_n}{\theta})\leqslant t\Big]=\mathbb{P}\Big[M_n \geqslant \frac{(n-t)\theta}{n} \Big] =1-\Big(\frac{n-t}{n}\Big)^{n} \rightarrow 1- e^{-t} \textbf{ as n} \rightarrow \infty\] 144 | 145 | Hence, $n(1-\frac{M_n}{\theta})$ converages in distribution to an exponential random variable with parameter 1. 146 | \item 3. 147 | Let A is an exponential random variable with parameter 1. Because $n(1-\frac{M_n}{\theta})$ converages in distribution to X, we have: 148 | \[\mathbb{P}\Big[ n(1-\frac{M_n}{\theta}) \leqslant t\Big] \rightarrow \mathbb{P}[X\leqslant t]= 1-e^{-t} \] 149 | $1-e^{-t}=0.95 \Rightarrow t=3$. We have: 150 | \[ \mathbb{P}\Big[ n(1-\frac{M_n}{\theta}) \leqslant 3\Big] \rightarrow 0.95\] 151 | which is: 152 | \[ \mathbb{P}\Big[ \theta \leqslant \frac{nM_n}{n-3}\Big] \rightarrow 0.95\] 153 | On the other hand, we always have $\theta \geqslant M_n$ (uniform distribution). Hence, we get: 154 | \[ \mathbb{P}\Big[M_n \leqslant \theta \leqslant \frac{nM_n}{n-3}\Big] \rightarrow 0.95 \textbf{ as n} \rightarrow \infty\] 155 | We conclude $L=\Big[M_n, \frac{nM_n}{n-3} \Big]=\Big[ M_n, M_n+ \frac{3M_n}{n-3} \Big]$. 156 | \item 4. 157 | $bias(M_n)=\mathbb{M_n}-\theta= \frac{n}{n+1}\theta -\theta \neq 0$. Hence, $M_n$ is biased. 158 | \end{problem} 159 | \end{document} -------------------------------------------------------------------------------- /PS3/ProblemSet3.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | -------------------------------------------------------------------------------- /PS3/ProblemSet3.log: -------------------------------------------------------------------------------- 1 | This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex 2019.9.8) 26 DEC 2019 18:58 2 | entering extended mode 3 | restricted \write18 enabled. 4 | %&-line parsing enabled. 5 | **ProblemSet3.tex 6 | (./ProblemSet3.tex 7 | LaTeX2e <2017-04-15> 8 | Babel <3.18> and hyphenation patterns for 84 language(s) loaded. 9 | (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls 10 | Document Class: article 2014/09/29 v1.4h Standard LaTeX document class 11 | (/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo 12 | File: size10.clo 2014/09/29 v1.4h Standard LaTeX file (size option) 13 | ) 14 | \c@part=\count79 15 | \c@section=\count80 16 | \c@subsection=\count81 17 | \c@subsubsection=\count82 18 | \c@paragraph=\count83 19 | \c@subparagraph=\count84 20 | \c@figure=\count85 21 | \c@table=\count86 22 | \abovecaptionskip=\skip41 23 | \belowcaptionskip=\skip42 24 | \bibindent=\dimen102 25 | ) 26 | (/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty 27 | Package: geometry 2010/09/12 v5.6 Page Geometry 28 | 29 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty 30 | Package: keyval 2014/10/28 v1.15 key=value parser (DPC) 31 | \KV@toks@=\toks14 32 | ) 33 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty 34 | Package: ifpdf 2017/03/15 v3.2 Provides the ifpdf switch 35 | ) 36 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifvtex.sty 37 | Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) 38 | Package ifvtex Info: VTeX not detected. 39 | ) 40 | (/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty 41 | Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional 42 | ) 43 | \Gm@cnth=\count87 44 | \Gm@cntv=\count88 45 | \c@Gm@tempcnt=\count89 46 | \Gm@bindingoffset=\dimen103 47 | \Gm@wd@mp=\dimen104 48 | \Gm@odd@mp=\dimen105 49 | \Gm@even@mp=\dimen106 50 | \Gm@layoutwidth=\dimen107 51 | \Gm@layoutheight=\dimen108 52 | \Gm@layouthoffset=\dimen109 53 | \Gm@layoutvoffset=\dimen110 54 | \Gm@dimlist=\toks15 55 | ) 56 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty 57 | Package: amsmath 2017/09/02 v2.17a AMS math features 58 | \@mathmargin=\skip43 59 | 60 | For additional information on amsmath, use the `?' option. 61 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty 62 | Package: amstext 2000/06/29 v2.01 AMS text 63 | 64 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty 65 | File: amsgen.sty 1999/11/30 v2.0 generic functions 66 | \@emptytoks=\toks16 67 | \ex@=\dimen111 68 | )) 69 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty 70 | Package: amsbsy 1999/11/29 v1.2d Bold Symbols 71 | \pmbraise@=\dimen112 72 | ) 73 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty 74 | Package: amsopn 2016/03/08 v2.02 operator names 75 | ) 76 | \inf@bad=\count90 77 | LaTeX Info: Redefining \frac on input line 213. 78 | \uproot@=\count91 79 | \leftroot@=\count92 80 | LaTeX Info: Redefining \overline on input line 375. 81 | \classnum@=\count93 82 | \DOTSCASE@=\count94 83 | LaTeX Info: Redefining \ldots on input line 472. 84 | LaTeX Info: Redefining \dots on input line 475. 85 | LaTeX Info: Redefining \cdots on input line 596. 86 | \Mathstrutbox@=\box26 87 | \strutbox@=\box27 88 | \big@size=\dimen113 89 | LaTeX Font Info: Redeclaring font encoding OML on input line 712. 90 | LaTeX Font Info: Redeclaring font encoding OMS on input line 713. 91 | \macc@depth=\count95 92 | \c@MaxMatrixCols=\count96 93 | \dotsspace@=\muskip10 94 | \c@parentequation=\count97 95 | \dspbrk@lvl=\count98 96 | \tag@help=\toks17 97 | \row@=\count99 98 | \column@=\count100 99 | \maxfields@=\count101 100 | \andhelp@=\toks18 101 | \eqnshift@=\dimen114 102 | \alignsep@=\dimen115 103 | \tagshift@=\dimen116 104 | \tagwidth@=\dimen117 105 | \totwidth@=\dimen118 106 | \lineht@=\dimen119 107 | \@envbody=\toks19 108 | \multlinegap=\skip44 109 | \multlinetaggap=\skip45 110 | \mathdisplay@stack=\toks20 111 | LaTeX Info: Redefining \[ on input line 2817. 112 | LaTeX Info: Redefining \] on input line 2818. 113 | ) 114 | (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty 115 | Package: amsthm 2017/10/31 v2.20.4 116 | \thm@style=\toks21 117 | \thm@bodyfont=\toks22 118 | \thm@headfont=\toks23 119 | \thm@notefont=\toks24 120 | \thm@headpunct=\toks25 121 | \thm@preskip=\skip46 122 | \thm@postskip=\skip47 123 | \thm@headsep=\skip48 124 | \dth@everypar=\toks26 125 | ) 126 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty 127 | Package: amssymb 2013/01/14 v3.01 AMS font symbols 128 | 129 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty 130 | Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support 131 | \symAMSa=\mathgroup4 132 | \symAMSb=\mathgroup5 133 | LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' 134 | (Font) U/euf/m/n --> U/euf/b/n on input line 106. 135 | )) 136 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty 137 | Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) 138 | 139 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty 140 | Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR) 141 | 142 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty 143 | Package: trig 2016/01/03 v1.10 sin cos tan (DPC) 144 | ) 145 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg 146 | File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration 147 | ) 148 | Package graphics Info: Driver file: pdftex.def on input line 99. 149 | 150 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def 151 | File: pdftex.def 2018/01/08 v1.0l Graphics/color driver for pdftex 152 | )) 153 | \Gin@req@height=\dimen120 154 | \Gin@req@width=\dimen121 155 | ) 156 | (/usr/share/texlive/texmf-dist/tex/latex/tools/multicol.sty 157 | Package: multicol 2017/04/11 v1.8q multicolumn formatting (FMi) 158 | \c@tracingmulticols=\count102 159 | \mult@box=\box28 160 | \multicol@leftmargin=\dimen122 161 | \c@unbalance=\count103 162 | \c@collectmore=\count104 163 | \doublecol@number=\count105 164 | \multicoltolerance=\count106 165 | \multicolpretolerance=\count107 166 | \full@width=\dimen123 167 | \page@free=\dimen124 168 | \premulticols=\dimen125 169 | \postmulticols=\dimen126 170 | \multicolsep=\skip49 171 | \multicolbaselineskip=\skip50 172 | \partial@page=\box29 173 | \last@line=\box30 174 | \maxbalancingoverflow=\dimen127 175 | \mult@rightbox=\box31 176 | \mult@grightbox=\box32 177 | \mult@gfirstbox=\box33 178 | \mult@firstbox=\box34 179 | \@tempa=\box35 180 | \@tempa=\box36 181 | \@tempa=\box37 182 | \@tempa=\box38 183 | \@tempa=\box39 184 | \@tempa=\box40 185 | \@tempa=\box41 186 | \@tempa=\box42 187 | \@tempa=\box43 188 | \@tempa=\box44 189 | \@tempa=\box45 190 | \@tempa=\box46 191 | \@tempa=\box47 192 | \@tempa=\box48 193 | \@tempa=\box49 194 | \@tempa=\box50 195 | \@tempa=\box51 196 | \c@columnbadness=\count108 197 | \c@finalcolumnbadness=\count109 198 | \last@try=\dimen128 199 | \multicolovershoot=\dimen129 200 | \multicolundershoot=\dimen130 201 | \mult@nat@firstbox=\box52 202 | \colbreak@box=\box53 203 | \mc@col@check@num=\count110 204 | ) 205 | (/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty 206 | Package: array 2016/10/06 v2.4d Tabular extension package (FMi) 207 | \col@sep=\dimen131 208 | \extrarowheight=\dimen132 209 | \NC@list=\toks27 210 | \extratabsurround=\skip51 211 | \backup@length=\skip52 212 | ) (./ProblemSet3.aux) 213 | \openout1 = `ProblemSet3.aux'. 214 | 215 | LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 14. 216 | LaTeX Font Info: ... okay on input line 14. 217 | LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 14. 218 | LaTeX Font Info: ... okay on input line 14. 219 | LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 14. 220 | LaTeX Font Info: ... okay on input line 14. 221 | LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 14. 222 | LaTeX Font Info: ... okay on input line 14. 223 | LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 14. 224 | LaTeX Font Info: ... okay on input line 14. 225 | LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 14. 226 | LaTeX Font Info: ... okay on input line 14. 227 | 228 | *geometry* driver: auto-detecting 229 | *geometry* detected driver: pdftex 230 | *geometry* verbose mode - [ preamble ] result: 231 | * driver: pdftex 232 | * paper: 233 | * layout: 234 | * layoutoffset:(h,v)=(0.0pt,0.0pt) 235 | * modes: 236 | * h-part:(L,W,R)=(72.26999pt, 469.75502pt, 72.26999pt) 237 | * v-part:(T,H,B)=(72.26999pt, 650.43001pt, 72.26999pt) 238 | * \paperwidth=614.295pt 239 | * \paperheight=794.96999pt 240 | * \textwidth=469.75502pt 241 | * \textheight=650.43001pt 242 | * \oddsidemargin=0.0pt 243 | * \evensidemargin=0.0pt 244 | * \topmargin=-37.0pt 245 | * \headheight=12.0pt 246 | * \headsep=25.0pt 247 | * \topskip=10.0pt 248 | * \footskip=30.0pt 249 | * \marginparwidth=65.0pt 250 | * \marginparsep=11.0pt 251 | * \columnsep=10.0pt 252 | * \skip\footins=9.0pt plus 4.0pt minus 2.0pt 253 | * \hoffset=0.0pt 254 | * \voffset=0.0pt 255 | * \mag=1000 256 | * \@twocolumnfalse 257 | * \@twosidefalse 258 | * \@mparswitchfalse 259 | * \@reversemarginfalse 260 | * (1in=72.27pt=25.4mm, 1cm=28.453pt) 261 | 262 | (/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii 263 | [Loading MPS to PDF converter (version 2006.09.02).] 264 | \scratchcounter=\count111 265 | \scratchdimen=\dimen133 266 | \scratchbox=\box54 267 | \nofMPsegments=\count112 268 | \nofMParguments=\count113 269 | \everyMPshowfont=\toks28 270 | \MPscratchCnt=\count114 271 | \MPscratchDim=\dimen134 272 | \MPnumerator=\count115 273 | \makeMPintoPDFobject=\count116 274 | \everyMPtoPDFconversion=\toks29 275 | ) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty 276 | Package: epstopdf-base 2016/05/15 v2.6 Base part for package epstopdf 277 | 278 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty 279 | Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) 280 | ) 281 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/grfext.sty 282 | Package: grfext 2016/05/16 v1.2 Manage graphics extensions (HO) 283 | 284 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvdefinekeys.sty 285 | Package: kvdefinekeys 2016/05/16 v1.4 Define keys (HO) 286 | 287 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty 288 | Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) 289 | ))) 290 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty 291 | Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) 292 | 293 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty 294 | Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) 295 | 296 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/etexcmds.sty 297 | Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) 298 | 299 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifluatex.sty 300 | Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) 301 | Package ifluatex Info: LuaTeX not detected. 302 | ) 303 | Package etexcmds Info: Could not find \expanded. 304 | (etexcmds) That can mean that you are not using pdfTeX 1.50 or 305 | (etexcmds) that some package has redefined \expanded. 306 | (etexcmds) In the latter case, load this package earlier. 307 | ))) 308 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/pdftexcmds.sty 309 | Package: pdftexcmds 2018/01/21 v0.26 Utility functions of pdfTeX for LuaTeX (HO 310 | ) 311 | Package pdftexcmds Info: LuaTeX not detected. 312 | Package pdftexcmds Info: \pdf@primitive is available. 313 | Package pdftexcmds Info: \pdf@ifprimitive is available. 314 | Package pdftexcmds Info: \pdfdraftmode found. 315 | ) 316 | Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 317 | 38. 318 | Package grfext Info: Graphics extension search list: 319 | (grfext) [.pdf,.png,.jpg,.mps,.jpeg,.jbig2,.jb2,.PDF,.PNG,.JPG,.JPE 320 | G,.JBIG2,.JB2,.eps] 321 | (grfext) \AppendGraphicsExtensions on input line 456. 322 | 323 | (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg 324 | File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv 325 | e 326 | )) 327 | LaTeX Font Info: Try loading font information for U+msa on input line 17. 328 | 329 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd 330 | File: umsa.fd 2013/01/14 v3.01 AMS symbols A 331 | ) 332 | LaTeX Font Info: Try loading font information for U+msb on input line 17. 333 | 334 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd 335 | File: umsb.fd 2013/01/14 v3.01 AMS symbols B 336 | ) 337 | Underfull \hbox (badness 10000) in paragraph at lines 46--47 338 | 339 | [] 340 | 341 | [1 342 | 343 | {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] 344 | Underfull \hbox (badness 10000) in paragraph at lines 53--54 345 | 346 | [] 347 | 348 | 349 | Underfull \hbox (badness 10000) in paragraph at lines 60--61 350 | 351 | [] 352 | 353 | 354 | Underfull \hbox (badness 10000) in paragraph at lines 62--69 355 | 356 | [] 357 | 358 | [2] 359 | Underfull \hbox (badness 10000) in paragraph at lines 84--86 360 | 361 | [] 362 | 363 | 364 | Underfull \hbox (badness 10000) in paragraph at lines 86--88 365 | 366 | [] 367 | 368 | 369 | Underfull \hbox (badness 10000) in paragraph at lines 93--95 370 | 371 | [] 372 | 373 | 374 | Underfull \hbox (badness 10000) in paragraph at lines 95--97 375 | 376 | [] 377 | 378 | [3] (./ProblemSet3.aux) ) 379 | Here is how much of TeX's memory you used: 380 | 3048 strings out of 492982 381 | 40879 string characters out of 6134897 382 | 118875 words of memory out of 5000000 383 | 6565 multiletter control sequences out of 15000+600000 384 | 11210 words of font info for 44 fonts, out of 8000000 for 9000 385 | 1141 hyphenation exceptions out of 8191 386 | 41i,11n,32p,327b,199s stack positions out of 5000i,500n,10000p,200000b,80000s 387 | 402 | Output written on ProblemSet3.pdf (3 pages, 150408 bytes). 403 | PDF statistics: 404 | 74 PDF objects out of 1000 (max. 8388607) 405 | 53 compressed objects within 1 object stream 406 | 0 named destinations out of 1000 (max. 500000) 407 | 1 words of extra memory for PDF output out of 10000 (max. 10000000) 408 | 409 | -------------------------------------------------------------------------------- /PS3/ProblemSet3.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS3/ProblemSet3.pdf -------------------------------------------------------------------------------- /PS3/ProblemSet3.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS3/ProblemSet3.synctex.gz -------------------------------------------------------------------------------- /PS3/ProblemSet3.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt]{article} 2 | 3 | \usepackage[margin=1in]{geometry} 4 | \usepackage{amsmath,amsthm,amssymb, graphicx, multicol, array} 5 | 6 | \newcommand{\N}{\mathbb{N}} 7 | \newcommand{\Z}{\mathbb{Z}} 8 | 9 | \newenvironment{problem}[2][Problem]{\begin{trivlist} 10 | \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} 11 | 12 | 13 | 14 | \begin{document} 15 | \title{Problem Set 3} 16 | \author{Hoang Nguyen, Huy Nguyen} 17 | \maketitle 18 | 19 | 20 | \begin{problem}{1} 21 | 22 | \item 1. 23 | We have : 24 | \[\ell_n=\sum_{i=1}^{n}(\log \theta + \theta\log\tau - (\theta+1)\log X_i) \mathbf{1}(X_i \geqslant \tau)\] 25 | \[\Rightarrow \frac{\partial \ell_n}{\partial \theta}=\sum_{i=1}^{n}(\frac{1}{\theta}+ \log \tau-\log X_i) \mathbf{1}(X_i \geqslant \tau)=0\] 26 | \[\Leftrightarrow \theta= \frac{\sum_{i=1}^{n}(\mathbf{1}(X_i \geqslant \tau))}{\sum_{i=1}^{n}(\log X_i-\log \tau)\mathbf{1}(X_i \geqslant \tau)}\] 27 | \item 3. 28 | \[\ell_n=\sum_{i=1}^{n}(\frac{1}{2}\log \theta+(\sqrt{\theta}-1)\log X_i)\mathbf{1}(0 \leqslant X_i \leqslant1)\] 29 | \[\Rightarrow \frac{\partial \ell_n}{\partial \theta}=(\frac{1}{2\theta}+\frac{\log X_i}{2\sqrt{\theta}})\mathbf{1}(0 \leqslant X_i \leqslant1)=0\] 30 | \[\Leftrightarrow \theta=\Big(\frac{\sum_{i=1}^{n}1}{\sum_{i=1}^{n}\log X_i}\Big)^2\] 31 | \item 4. 32 | \[\ell_n=\sum_{i=1}^{n}(\log X_i-2\log \theta-\frac{{X_i}^2}{2\theta^2})\mathbf{1}(X_i \geqslant 0)\] 33 | \[\Leftrightarrow \frac{\partial \ell_n}{\partial \theta}=\sum_{i=1}^{n}(-\frac{2}{\theta}+\frac{{X_i}^2}{\theta^3})\mathbf{1}(X_i \geqslant 0)=0\] 34 | \[\Leftrightarrow \theta=\frac{\sum_{i=1}^{n}(X_i)\mathbf{1}(X_i \geqslant 0)}{\sum_{i=1}^{n}\sqrt{2}\mathbf{1}(X_i \geqslant 0)}\] 35 | \item 5. 36 | \[\ell_n=\sum_{i=1}^{n}(\log \theta+ \log \tau+(\tau-1)\log X_i-\theta{X_i}^{\tau})\mathbf{1}(X_i \geqslant 0)\] 37 | \[\Leftrightarrow \frac{\partial \ell_n}{\partial \theta}=\sum_{i=1}^{n}(\frac{1}{\theta}-{X_i}^{\tau})\mathbf{1}(X_i \geqslant 0)\] 38 | \[\Leftrightarrow \theta=\frac{\sum_{i=1}^{n} 1\mathbf{1}(X_i \geqslant 0) }{\sum_{i=1}^{n} ({X_i}^{\tau})\mathbf{1}(X_i \geqslant 0)}\] 39 | 40 | 41 | 42 | \end{problem} 43 | 44 | \begin{problem}{2} 45 | 46 | We have :\\ 47 | \[\mathcal{L}(\mu, \sigma)=\prod_{i=1}^n\Big(\frac{1}{\sigma}\exp({-\frac{1}{2\sigma^2}(X_i-\mu)^2})\Big)\] 48 | \[\ell_n=\log \mathcal{L}(\mu, \sigma)=\sum_{i=1}^n\Big( -\log\sigma-\frac{1}{2\sigma^2}(X_i-\mu)^2\Big)\] 49 | \[=\sum_{i=1}^n\Big(-\log\sigma -\frac{1}{2\sigma^2}(X_i-\bar{X_n}+\bar{X_n}-\mu)^2\Big)\] 50 | \[=\sum_{i=1}^n\Big(-\log\sigma -\frac{1}{2\sigma^2}\Big((X_i-\bar{X})^2+2(X_i-\bar{X})(\bar{X}-\mu) +(\bar{X}-\mu)^2\Big)\Big)\] 51 | \[-n\log\sigma - \frac{1}{2\sigma^2}\Big( \sum_{i=1}^n(X_i-\bar{X})^2+2(\bar{X}-\mu)\sum_{i=1}^n(X_i-\bar{X})+n(\bar{X}-\mu)^2\Big)\] 52 | 53 | Because $\sum_{i=1}^n(X_i-\bar{X}=0)$, we get : \\ 54 | \[\ell_n=-n\log\sigma -\frac{\sum_{i=1}^n(X_i-\bar{X})^2}{2\sigma^2}-\frac{n(\bar{X}-\mu)}{2\sigma^2}\] 55 | 56 | \[\Rightarrow \begin{cases} \frac{\partial \ell_n}{\partial \mu}=\frac{n(\bar{X 57 | }-\mu)}{\sigma^2}=0\\ \frac{\partial \ell_n}{\partial \sigma}=\frac{-n}{\sigma}+\frac{1}{\sigma^3}\Big( \sum_{i=n}^n(X_i-\bar{X_n})^2+n(\bar{X_n}-\mu)^2\Big)=0 \end{cases}\] 58 | \[\Leftrightarrow \begin{cases} \mu=\bar{X_n}\\ \sigma^2=\frac{\sum_{i=1}^n(X_i-\bar{X_n})^2}{n}\end{cases}\] 59 | 60 | Prove this this is consitent: \\ 61 | 62 | To do this, we need to prove bias $\rightarrow$ 0 and $se \rightarrow 0$ as n $\rightarrow \infty$.\\ 63 | $\bullet$ Prove $\mu_{MLE}$ is consistent: 64 | $\mathbb{E}[\bar{X_n}]=\mathbb{E}[\frac{1}{n}\sum_{i=1}^{n}X_i]=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[X_i]=\frac{1}{n}\sum_{i=1}^{n}\mu=\mu$. Hence, $bias(\mu_{MLE})=\mathbb{E}[\mu_{MLE}]-\mu=\mathbb{E}[\bar{X_n}]-\mu=0$. (1)\\ 65 | $\mathbb{V}[\bar{X_n}]=\mathbb{V}[\frac{1}{n}\sum_{i=1}^{n}X_i]=\frac{1}{n^2}\mathbb{V}[\sum_{i=1}^{n}X_i]=\frac{\sigma^2}{n} \longrightarrow 0$ as $n \longrightarrow \infty$. Hence, $se(\mu_{MLE})=se(\bar{X_n})=\sqrt{\mathbb{V}[\bar{X_n}]} \longrightarrow 0$ as $n \longrightarrow \infty$. (2)\\ 66 | Using (1) and (2), we conclude that $\mu_{MLE}$ is consistent.\\ 67 | $\bullet$ Prove $\sigma_{MLE}^2$ is consistent:\\ 68 | We have $\mathbb{E}[\sigma_{MLE}^2]=\mathbb{E}\Big[\sum_{i=1}^{n}\frac{(X_i-\bar{X_n})^2}{n} \Big]\Rightarrow n\mathbb{E}[\sigma_{MLE}^2]=\mathbb{E}\Big[\sum_{i=1}^{n}(X_i-\bar{X_n})^2 \Big]$\\ 69 | \[n\mathbb{E}[\sigma_{MLE}^2]=\mathbb{E}\Big[\sum_{i=1}^{n}(X_i-\bar{X_n})^2 \Big]=\mathbb{E}\Big[\sum_{i=1}^{n}((X_i-\mu)+(\mu+\bar{X_n}))^2 \Big] \] 70 | \[=\mathbb{E}\Big[ \sum_{i=1}^{n}(X_i-\mu)^2 + \sum_{i=1}^{n}(\mu-\bar{X_n})^2 + 2\sum_{i=1}^{n}(X_i-\mu)(\mu-\bar{X_n}) \Big] \] 71 | \[=\mathbb{E}\Big[ \sum_{i=1}^{n}(X_i-\mu)^2 +n(\mu-\bar{X_n})^2 + 2(\mu-\bar{X_n})\sum_{i=1}^{n}(X_i-\mu) \Big] \] 72 | \[=\mathbb{E}\Big[ \sum_{i=1}^{n}(X_i-\mu)^2 +n(\mu-\bar{X_n})^2 +2(\mu-\bar{X_n})n(\bar{X_n}-\mu) \Big] \] 73 | \[=\mathbb{E}\Big[\sum_{i=1}^{n}(X_i-\mu)^2-n(\mu-\bar{X_n})^2 \Big]=\sum_{i=1}^{n}\mathbb{E}[(X_i-\mu)^2]-n\mathbb{E}[(\mu-\bar{X_n})^2] \] 74 | \[=n\sigma^2-n\mathbb{V}[\bar{X_n}]=n\sigma^2-n\frac{\sigma^2}{n}=n\sigma^2-\sigma^2=(n-1)\sigma^2 \] 75 | So, $\mathbb{E}[\sigma_{MLE}^2]=\frac{n-1}{n}\sigma^2 \longrightarrow 0$ as $n \longrightarrow \infty$. (3)\\ 76 | We have: $\frac{n\sigma_{MLE}^{2}}{\sigma^2} \sim \chi_{n-1}^2$. Hence $\mathbb{V}[\frac{n\sigma_{MLE}^{2}}{\sigma^2}]=\mathbb{V}[\chi_{n-1}^2]=2(n-1)$, which equivalents to $\mathbb{V}[\sigma_{MLE}^2]=\frac{2(n-1)\sigma^4}{n^2}$. This actually tends to 0 as n tends to infinity. (4)\\ 77 | Using (3) and (4), we conclude that $\sigma_{MLE}^2$ is consistent. 78 | 79 | 80 | 81 | 82 | \end{problem} 83 | \begin{problem}{3} 84 | \item 1. 85 | We have: $KL(\mathcal{N}(a, \sigma^2), \mathcal{N}(b, \sigma^2))=\mathbb{E}_P\left[\log{\frac{f_P(x)}{f_Q(x)}}\right]=\mathbb{E}_P\left[\log{f_P(x)}\right] - \mathbb{E}_P\left[\log{f_Q(x)}\right] = \frac{(a-b)^2}{2\sigma^2}$.\\ 86 | \item 2. 87 | We have: $KL(Ber(a), Ber(b))=\mathbb{E}_P\left[\log{\frac{f_P(x)}{f_Q(x)}}\right]=\sum_{x \in E}p(x)\log{\frac{p(x)}{q(x)}} = a\log{\frac{a}{b}} + (1-a)\log{\frac{1-a}{1-b}}$.\\ 88 | %%% expectations involve logarithms 89 | %%% expectations computed with respect to the P measure, not the Q measure 90 | \end{problem} 91 | 92 | 93 | \begin{problem}{4} 94 | \item 1. 95 | We have: $TV(Uni(0,s), Uni(0,t))=\frac{1}{2}\int_{0}^{s}|\frac{1}{s}-\frac{1}{t}|dx + \frac{1}{2}\int_{s}^{t}|-\frac{1}{t}|dx=\frac{t-s}{t}$.\\ 96 | \item 2. 97 | We have: $TV(Ber(p), Ber(q))=\frac{1}{2}\sum_{i=0}^{1}|Ber(i|p)-Ber(i|q)|=\frac{1}{2}(|p-q|+|1-p-(1-q)|)=|p-q|$.\\ 98 | \item 3. 99 | From previous question, we get: $TV(Ber(\bar{X_n}), Ber(p))=|\bar{X_n}-p|$. On the other hand, we know $\bar{X_n}$ converges to $p$ in probability because $\mathbb{P}[|\bar{X_n}-p|>\epsilon] \longrightarrow 0$ (Law of large number). Hence, $Ber(\bar{X_n})$ and $Ber(p)$ converges to zero in probability. 100 | \item 4. I wonder what Dirac distribution is?? 101 | \end{problem} 102 | 103 | \end{document} 104 | -------------------------------------------------------------------------------- /PS4/ProbemSet4.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 4 | \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined 5 | \global\let\oldcontentsline\contentsline 6 | \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} 7 | \global\let\oldnewlabel\newlabel 8 | \gdef\newlabel#1#2{\newlabelxx{#1}#2} 9 | \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} 10 | \AtEndDocument{\ifx\hyper@anchor\@undefined 11 | \let\contentsline\oldcontentsline 12 | \let\newlabel\oldnewlabel 13 | \fi} 14 | \fi} 15 | \global\let\hyper@last\relax 16 | \gdef\HyperFirstAtBeginDocument#1{#1} 17 | \providecommand\HyField@AuxAddToFields[1]{} 18 | \providecommand\HyField@AuxAddToCoFields[2]{} 19 | -------------------------------------------------------------------------------- /PS4/ProbemSet4.log: -------------------------------------------------------------------------------- 1 | This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex 2019.9.8) 29 DEC 2019 13:37 2 | entering extended mode 3 | restricted \write18 enabled. 4 | %&-line parsing enabled. 5 | **ProbemSet4.tex 6 | (./ProbemSet4.tex 7 | LaTeX2e <2017-04-15> 8 | Babel <3.18> and hyphenation patterns for 84 language(s) loaded. 9 | (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls 10 | Document Class: article 2014/09/29 v1.4h Standard LaTeX document class 11 | (/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo 12 | File: size10.clo 2014/09/29 v1.4h Standard LaTeX file (size option) 13 | ) 14 | \c@part=\count79 15 | \c@section=\count80 16 | \c@subsection=\count81 17 | \c@subsubsection=\count82 18 | \c@paragraph=\count83 19 | \c@subparagraph=\count84 20 | \c@figure=\count85 21 | \c@table=\count86 22 | \abovecaptionskip=\skip41 23 | \belowcaptionskip=\skip42 24 | \bibindent=\dimen102 25 | ) 26 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty 27 | Package: hyperref 2018/02/06 v6.86b Hypertext links for LaTeX 28 | 29 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty 30 | Package: hobsub-hyperref 2016/05/16 v1.14 Bundle oberdiek, subset hyperref (HO) 31 | 32 | 33 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty 34 | Package: hobsub-generic 2016/05/16 v1.14 Bundle oberdiek, subset generic (HO) 35 | Package: hobsub 2016/05/16 v1.14 Construct package bundles (HO) 36 | Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) 37 | Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) 38 | Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) 39 | Package ifluatex Info: LuaTeX not detected. 40 | Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) 41 | Package ifvtex Info: VTeX not detected. 42 | Package: intcalc 2016/05/16 v1.2 Expandable calculations with integers (HO) 43 | Package: ifpdf 2017/03/15 v3.2 Provides the ifpdf switch 44 | Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) 45 | Package etexcmds Info: Could not find \expanded. 46 | (etexcmds) That can mean that you are not using pdfTeX 1.50 or 47 | (etexcmds) that some package has redefined \expanded. 48 | (etexcmds) In the latter case, load this package earlier. 49 | Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) 50 | Package: kvdefinekeys 2016/05/16 v1.4 Define keys (HO) 51 | Package: pdftexcmds 2018/01/21 v0.26 Utility functions of pdfTeX for LuaTeX (HO 52 | ) 53 | Package pdftexcmds Info: LuaTeX not detected. 54 | Package pdftexcmds Info: \pdf@primitive is available. 55 | Package pdftexcmds Info: \pdf@ifprimitive is available. 56 | Package pdftexcmds Info: \pdfdraftmode found. 57 | Package: pdfescape 2016/05/16 v1.14 Implements pdfTeX's escape features (HO) 58 | Package: bigintcalc 2016/05/16 v1.4 Expandable calculations on big integers (HO 59 | ) 60 | Package: bitset 2016/05/16 v1.2 Handle bit-vector datatype (HO) 61 | Package: uniquecounter 2016/05/16 v1.3 Provide unlimited unique counter (HO) 62 | ) 63 | Package hobsub Info: Skipping package `hobsub' (already loaded). 64 | Package: letltxmacro 2016/05/16 v1.5 Let assignment for LaTeX macros (HO) 65 | Package: hopatch 2016/05/16 v1.3 Wrapper for package hooks (HO) 66 | Package: xcolor-patch 2016/05/16 xcolor patch 67 | Package: atveryend 2016/05/16 v1.9 Hooks at the very end of document (HO) 68 | Package atveryend Info: \enddocument detected (standard20110627). 69 | Package: atbegshi 2016/06/09 v1.18 At begin shipout hook (HO) 70 | Package: refcount 2016/05/16 v3.5 Data extraction from label references (HO) 71 | Package: hycolor 2016/05/16 v1.8 Color options for hyperref/bookmark (HO) 72 | ) 73 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty 74 | Package: keyval 2014/10/28 v1.15 key=value parser (DPC) 75 | \KV@toks@=\toks14 76 | ) 77 | (/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty 78 | Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional 79 | ) 80 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty 81 | Package: auxhook 2016/05/16 v1.4 Hooks for auxiliary files (HO) 82 | ) 83 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty 84 | Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) 85 | ) 86 | \@linkdim=\dimen103 87 | \Hy@linkcounter=\count87 88 | \Hy@pagecounter=\count88 89 | 90 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def 91 | File: pd1enc.def 2018/02/06 v6.86b Hyperref: PDFDocEncoding definition (HO) 92 | ) 93 | \Hy@SavedSpaceFactor=\count89 94 | 95 | (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg 96 | File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive 97 | ) 98 | Package hyperref Info: Hyper figures OFF on input line 4509. 99 | Package hyperref Info: Link nesting OFF on input line 4514. 100 | Package hyperref Info: Hyper index ON on input line 4517. 101 | Package hyperref Info: Plain pages OFF on input line 4524. 102 | Package hyperref Info: Backreferencing OFF on input line 4529. 103 | Package hyperref Info: Implicit mode ON; LaTeX internals redefined. 104 | Package hyperref Info: Bookmarks ON on input line 4762. 105 | \c@Hy@tempcnt=\count90 106 | 107 | (/usr/share/texlive/texmf-dist/tex/latex/url/url.sty 108 | \Urlmuskip=\muskip10 109 | Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. 110 | ) 111 | LaTeX Info: Redefining \url on input line 5115. 112 | \XeTeXLinkMargin=\dimen104 113 | \Fld@menulength=\count91 114 | \Field@Width=\dimen105 115 | \Fld@charsize=\dimen106 116 | Package hyperref Info: Hyper figures OFF on input line 6369. 117 | Package hyperref Info: Link nesting OFF on input line 6374. 118 | Package hyperref Info: Hyper index ON on input line 6377. 119 | Package hyperref Info: backreferencing OFF on input line 6384. 120 | Package hyperref Info: Link coloring OFF on input line 6389. 121 | Package hyperref Info: Link coloring with OCG OFF on input line 6394. 122 | Package hyperref Info: PDF/A mode OFF on input line 6399. 123 | LaTeX Info: Redefining \ref on input line 6439. 124 | LaTeX Info: Redefining \pageref on input line 6443. 125 | \Hy@abspage=\count92 126 | \c@Item=\count93 127 | \c@Hfootnote=\count94 128 | ) 129 | Package hyperref Info: Driver (autodetected): hpdftex. 130 | 131 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def 132 | File: hpdftex.def 2018/02/06 v6.86b Hyperref driver for pdfTeX 133 | \Fld@listcount=\count95 134 | \c@bookmark@seq@number=\count96 135 | 136 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty 137 | Package: rerunfilecheck 2016/05/16 v1.8 Rerun checks for auxiliary files (HO) 138 | Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 139 | 82. 140 | ) 141 | \Hy@SectionHShift=\skip43 142 | ) 143 | (/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty 144 | Package: geometry 2010/09/12 v5.6 Page Geometry 145 | \Gm@cnth=\count97 146 | \Gm@cntv=\count98 147 | \c@Gm@tempcnt=\count99 148 | \Gm@bindingoffset=\dimen107 149 | \Gm@wd@mp=\dimen108 150 | \Gm@odd@mp=\dimen109 151 | \Gm@even@mp=\dimen110 152 | \Gm@layoutwidth=\dimen111 153 | \Gm@layoutheight=\dimen112 154 | \Gm@layouthoffset=\dimen113 155 | \Gm@layoutvoffset=\dimen114 156 | \Gm@dimlist=\toks15 157 | ) 158 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty 159 | Package: amsmath 2017/09/02 v2.17a AMS math features 160 | \@mathmargin=\skip44 161 | 162 | For additional information on amsmath, use the `?' option. 163 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty 164 | Package: amstext 2000/06/29 v2.01 AMS text 165 | 166 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty 167 | File: amsgen.sty 1999/11/30 v2.0 generic functions 168 | \@emptytoks=\toks16 169 | \ex@=\dimen115 170 | )) 171 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty 172 | Package: amsbsy 1999/11/29 v1.2d Bold Symbols 173 | \pmbraise@=\dimen116 174 | ) 175 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty 176 | Package: amsopn 2016/03/08 v2.02 operator names 177 | ) 178 | \inf@bad=\count100 179 | LaTeX Info: Redefining \frac on input line 213. 180 | \uproot@=\count101 181 | \leftroot@=\count102 182 | LaTeX Info: Redefining \overline on input line 375. 183 | \classnum@=\count103 184 | \DOTSCASE@=\count104 185 | LaTeX Info: Redefining \ldots on input line 472. 186 | LaTeX Info: Redefining \dots on input line 475. 187 | LaTeX Info: Redefining \cdots on input line 596. 188 | \Mathstrutbox@=\box26 189 | \strutbox@=\box27 190 | \big@size=\dimen117 191 | LaTeX Font Info: Redeclaring font encoding OML on input line 712. 192 | LaTeX Font Info: Redeclaring font encoding OMS on input line 713. 193 | \macc@depth=\count105 194 | \c@MaxMatrixCols=\count106 195 | \dotsspace@=\muskip11 196 | \c@parentequation=\count107 197 | \dspbrk@lvl=\count108 198 | \tag@help=\toks17 199 | \row@=\count109 200 | \column@=\count110 201 | \maxfields@=\count111 202 | \andhelp@=\toks18 203 | \eqnshift@=\dimen118 204 | \alignsep@=\dimen119 205 | \tagshift@=\dimen120 206 | \tagwidth@=\dimen121 207 | \totwidth@=\dimen122 208 | \lineht@=\dimen123 209 | \@envbody=\toks19 210 | \multlinegap=\skip45 211 | \multlinetaggap=\skip46 212 | \mathdisplay@stack=\toks20 213 | LaTeX Info: Redefining \[ on input line 2817. 214 | LaTeX Info: Redefining \] on input line 2818. 215 | ) 216 | (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty 217 | Package: amsthm 2017/10/31 v2.20.4 218 | \thm@style=\toks21 219 | \thm@bodyfont=\toks22 220 | \thm@headfont=\toks23 221 | \thm@notefont=\toks24 222 | \thm@headpunct=\toks25 223 | \thm@preskip=\skip47 224 | \thm@postskip=\skip48 225 | \thm@headsep=\skip49 226 | \dth@everypar=\toks26 227 | ) 228 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty 229 | Package: amssymb 2013/01/14 v3.01 AMS font symbols 230 | 231 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty 232 | Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support 233 | \symAMSa=\mathgroup4 234 | \symAMSb=\mathgroup5 235 | LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' 236 | (Font) U/euf/m/n --> U/euf/b/n on input line 106. 237 | )) 238 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty 239 | Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) 240 | 241 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty 242 | Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR) 243 | 244 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty 245 | Package: trig 2016/01/03 v1.10 sin cos tan (DPC) 246 | ) 247 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg 248 | File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration 249 | ) 250 | Package graphics Info: Driver file: pdftex.def on input line 99. 251 | 252 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def 253 | File: pdftex.def 2018/01/08 v1.0l Graphics/color driver for pdftex 254 | )) 255 | \Gin@req@height=\dimen124 256 | \Gin@req@width=\dimen125 257 | ) 258 | (/usr/share/texlive/texmf-dist/tex/latex/tools/multicol.sty 259 | Package: multicol 2017/04/11 v1.8q multicolumn formatting (FMi) 260 | \c@tracingmulticols=\count112 261 | \mult@box=\box28 262 | \multicol@leftmargin=\dimen126 263 | \c@unbalance=\count113 264 | \c@collectmore=\count114 265 | \doublecol@number=\count115 266 | \multicoltolerance=\count116 267 | \multicolpretolerance=\count117 268 | \full@width=\dimen127 269 | \page@free=\dimen128 270 | \premulticols=\dimen129 271 | \postmulticols=\dimen130 272 | \multicolsep=\skip50 273 | \multicolbaselineskip=\skip51 274 | \partial@page=\box29 275 | \last@line=\box30 276 | \maxbalancingoverflow=\dimen131 277 | \mult@rightbox=\box31 278 | \mult@grightbox=\box32 279 | \mult@gfirstbox=\box33 280 | \mult@firstbox=\box34 281 | \@tempa=\box35 282 | \@tempa=\box36 283 | \@tempa=\box37 284 | \@tempa=\box38 285 | \@tempa=\box39 286 | \@tempa=\box40 287 | \@tempa=\box41 288 | \@tempa=\box42 289 | \@tempa=\box43 290 | \@tempa=\box44 291 | \@tempa=\box45 292 | \@tempa=\box46 293 | \@tempa=\box47 294 | \@tempa=\box48 295 | \@tempa=\box49 296 | \@tempa=\box50 297 | \@tempa=\box51 298 | \c@columnbadness=\count118 299 | \c@finalcolumnbadness=\count119 300 | \last@try=\dimen132 301 | \multicolovershoot=\dimen133 302 | \multicolundershoot=\dimen134 303 | \mult@nat@firstbox=\box52 304 | \colbreak@box=\box53 305 | \mc@col@check@num=\count120 306 | ) 307 | (/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty 308 | Package: array 2016/10/06 v2.4d Tabular extension package (FMi) 309 | \col@sep=\dimen135 310 | \extrarowheight=\dimen136 311 | \NC@list=\toks27 312 | \extratabsurround=\skip52 313 | \backup@length=\skip53 314 | ) 315 | (/usr/share/texlive/texmf-dist/tex/latex/bbold/bbold.sty 316 | Package: bbold 1994/04/06 Bbold symbol package 317 | LaTeX Font Info: Redeclaring math alphabet \mathbb on input line 42. 318 | ) (./ProbemSet4.aux) 319 | \openout1 = `ProbemSet4.aux'. 320 | 321 | LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 14. 322 | LaTeX Font Info: ... okay on input line 14. 323 | LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 14. 324 | LaTeX Font Info: ... okay on input line 14. 325 | LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 14. 326 | LaTeX Font Info: ... okay on input line 14. 327 | LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 14. 328 | LaTeX Font Info: ... okay on input line 14. 329 | LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 14. 330 | LaTeX Font Info: ... okay on input line 14. 331 | LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 14. 332 | LaTeX Font Info: ... okay on input line 14. 333 | LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 14. 334 | LaTeX Font Info: ... okay on input line 14. 335 | \AtBeginShipoutBox=\box54 336 | Package hyperref Info: Link coloring OFF on input line 14. 337 | 338 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty 339 | Package: nameref 2016/05/21 v2.44 Cross-referencing by name of section 340 | 341 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty 342 | Package: gettitlestring 2016/05/16 v1.5 Cleanup title references (HO) 343 | ) 344 | \c@section@level=\count121 345 | ) 346 | LaTeX Info: Redefining \ref on input line 14. 347 | LaTeX Info: Redefining \pageref on input line 14. 348 | LaTeX Info: Redefining \nameref on input line 14. 349 | 350 | (./ProbemSet4.out) (./ProbemSet4.out) 351 | \@outlinefile=\write3 352 | \openout3 = `ProbemSet4.out'. 353 | 354 | 355 | *geometry* driver: auto-detecting 356 | *geometry* detected driver: pdftex 357 | *geometry* verbose mode - [ preamble ] result: 358 | * driver: pdftex 359 | * paper: 360 | * layout: 361 | * layoutoffset:(h,v)=(0.0pt,0.0pt) 362 | * modes: 363 | * h-part:(L,W,R)=(72.26999pt, 469.75502pt, 72.26999pt) 364 | * v-part:(T,H,B)=(72.26999pt, 650.43001pt, 72.26999pt) 365 | * \paperwidth=614.295pt 366 | * \paperheight=794.96999pt 367 | * \textwidth=469.75502pt 368 | * \textheight=650.43001pt 369 | * \oddsidemargin=0.0pt 370 | * \evensidemargin=0.0pt 371 | * \topmargin=-37.0pt 372 | * \headheight=12.0pt 373 | * \headsep=25.0pt 374 | * \topskip=10.0pt 375 | * \footskip=30.0pt 376 | * \marginparwidth=65.0pt 377 | * \marginparsep=11.0pt 378 | * \columnsep=10.0pt 379 | * \skip\footins=9.0pt plus 4.0pt minus 2.0pt 380 | * \hoffset=0.0pt 381 | * \voffset=0.0pt 382 | * \mag=1000 383 | * \@twocolumnfalse 384 | * \@twosidefalse 385 | * \@mparswitchfalse 386 | * \@reversemarginfalse 387 | * (1in=72.27pt=25.4mm, 1cm=28.453pt) 388 | 389 | (/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii 390 | [Loading MPS to PDF converter (version 2006.09.02).] 391 | \scratchcounter=\count122 392 | \scratchdimen=\dimen137 393 | \scratchbox=\box55 394 | \nofMPsegments=\count123 395 | \nofMParguments=\count124 396 | \everyMPshowfont=\toks28 397 | \MPscratchCnt=\count125 398 | \MPscratchDim=\dimen138 399 | \MPnumerator=\count126 400 | \makeMPintoPDFobject=\count127 401 | \everyMPtoPDFconversion=\toks29 402 | ) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty 403 | Package: epstopdf-base 2016/05/15 v2.6 Base part for package epstopdf 404 | 405 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/grfext.sty 406 | Package: grfext 2016/05/16 v1.2 Manage graphics extensions (HO) 407 | ) 408 | Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 409 | 38. 410 | Package grfext Info: Graphics extension search list: 411 | (grfext) [.pdf,.png,.jpg,.mps,.jpeg,.jbig2,.jb2,.PDF,.PNG,.JPG,.JPE 412 | G,.JBIG2,.JB2,.eps] 413 | (grfext) \AppendGraphicsExtensions on input line 456. 414 | 415 | (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg 416 | File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv 417 | e 418 | )) 419 | LaTeX Font Info: Try loading font information for U+msa on input line 18. 420 | 421 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd 422 | File: umsa.fd 2013/01/14 v3.01 AMS symbols A 423 | ) 424 | LaTeX Font Info: Try loading font information for U+msb on input line 18. 425 | 426 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd 427 | File: umsb.fd 2013/01/14 v3.01 AMS symbols B 428 | ) 429 | LaTeX Font Info: Try loading font information for U+bbold on input line 27. 430 | 431 | (/usr/share/texlive/texmf-dist/tex/latex/bbold/Ubbold.fd) 432 | Underfull \hbox (badness 10000) in paragraph at lines 35--38 433 | 434 | [] 435 | 436 | 437 | Underfull \hbox (badness 10000) in paragraph at lines 45--48 438 | 439 | [] 440 | 441 | [1 442 | 443 | {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] 444 | Overfull \hbox (0.8728pt too wide) detected at line 88 445 | \OML/cmm/m/it/10 `[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^V; ^^[[]\OT1/cmr/m/n/10 446 | ) = \OML/cmm/m/it/10 log\OT1/cmr/m/n/10 ([] []\OML/cmm/m/it/10 e[]\OT1/cmr/m/n 447 | /10 ) = \OMS/cmsy/m/n/10 ^^@[]\OML/cmm/m/it/10 log\OT1/cmr/m/n/10 (2\OML/cmm/m/ 448 | it/10 ^^Y\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ []\OML/cmm/m/it/10 log\OT1/cmr/ 449 | m/n/10 (\OML/cmm/m/it/10 ^^[[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ [] \OML/cm 450 | m/m/it/10 log\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m 451 | /n/10 ^^@ [] []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 log\OT1/cmr/m/n/10 (\OML/cmm/m 452 | /it/10 X[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 ^^V\OT1/cmr/m 453 | /n/10 )[] 454 | [] 455 | 456 | 457 | Underfull \hbox (badness 10000) in paragraph at lines 93--95 458 | 459 | [] 460 | 461 | [2] 462 | Underfull \hbox (badness 10000) in paragraph at lines 112--115 463 | 464 | [] 465 | 466 | 467 | Underfull \hbox (badness 10000) in paragraph at lines 130--131 468 | 469 | [] 470 | 471 | 472 | Underfull \hbox (badness 10000) in paragraph at lines 131--132 473 | 474 | [] 475 | 476 | 477 | Underfull \hbox (badness 10000) in paragraph at lines 132--133 478 | 479 | [] 480 | 481 | [3] 482 | Package atveryend Info: Empty hook `BeforeClearDocument' on input line 174. 483 | [4] 484 | Package atveryend Info: Empty hook `AfterLastShipout' on input line 174. 485 | (./ProbemSet4.aux) 486 | Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 174. 487 | Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 174. 488 | Package rerunfilecheck Info: File `ProbemSet4.out' has not changed. 489 | (rerunfilecheck) Checksum: D41D8CD98F00B204E9800998ECF8427E;0. 490 | Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 174. 491 | ) 492 | Here is how much of TeX's memory you used: 493 | 6569 strings out of 492982 494 | 94267 string characters out of 6134897 495 | 201287 words of memory out of 5000000 496 | 10061 multiletter control sequences out of 15000+600000 497 | 11152 words of font info for 45 fonts, out of 8000000 for 9000 498 | 1141 hyphenation exceptions out of 8191 499 | 28i,10n,32p,680b,391s stack positions out of 5000i,500n,10000p,200000b,80000s 500 | < 512 | /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb> 516 | Output written on ProbemSet4.pdf (4 pages, 170711 bytes). 517 | PDF statistics: 518 | 93 PDF objects out of 1000 (max. 8388607) 519 | 69 compressed objects within 1 object stream 520 | 5 named destinations out of 1000 (max. 500000) 521 | 1 words of extra memory for PDF output out of 10000 (max. 10000000) 522 | 523 | -------------------------------------------------------------------------------- /PS4/ProbemSet4.out: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS4/ProbemSet4.out -------------------------------------------------------------------------------- /PS4/ProbemSet4.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS4/ProbemSet4.pdf -------------------------------------------------------------------------------- /PS4/ProbemSet4.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS4/ProbemSet4.synctex.gz -------------------------------------------------------------------------------- /PS4/ProbemSet4.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt]{article} 2 | \usepackage{hyperref} 3 | \usepackage[margin=1in]{geometry} 4 | \usepackage{amsmath,amsthm,amssymb, graphicx, multicol, array} 5 | \usepackage{bbold} 6 | \newcommand{\N}{\mathbb{N}} 7 | \newcommand{\Z}{\mathbb{Z}} 8 | 9 | \newenvironment{problem}[2][Problem]{\begin{trivlist} 10 | \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} 11 | 12 | 13 | 14 | \begin{document} 15 | \title{Problem Set 4} 16 | \author{Hoang Nguyen, Huy Nguyen} 17 | \maketitle 18 | 19 | \begin{problem}{1} 20 | \item 1. 21 | We have the log-likelihood function: 22 | \[\ell_n(p) = log(\prod_{i=1}^{n}p^{X_i}(1-p)^{1-X_i})=log(p)\sum_{i=1}^{n}X_i + log(1-p)\sum_{i=1}^{n}(1-X_i) \] 23 | Hence, 24 | \[\frac{\partial\ell_n(p)}{\partial p}=\frac{\sum_{i=1}^{n}X_i}{p} - \frac{\sum_{i=1}^{n}(1 - X_i)}{1-p} \] 25 | Set this equals to 0, we get: $p=\frac{1}{n}\sum_{i=1}^{n}X_i$.\\ 26 | We have: 27 | \[I(\theta) =\mathbb{E}[(\frac{\partial\ell_n(p)}{\partial p})^2]=\mathbb{E}\Big[(\frac{X}{p} - \frac{1-X}{1-p})^2 \Big] = \mathbb{E}[\frac{X^2}{p^2}] - 2\mathbb{E}[\frac{X - X^2}{p(1-p)}] + \mathbb{E}[\frac{X^2-2X+1}{(1-p)^2}] \] 28 | From Bernoulli distribution, we know: $\mathbb{E}[X]=p$ and $\mathbb{E}[X^2]=p$. Plugging those thing to equation above, we get: 29 | \[ I(\theta)=\frac{p}{p^2} - 2\frac{0-0}{p(1-p)} + \frac{p-2p+1}{(1-p)^2}=\frac{1}{p(1-p)} \] 30 | 31 | \item 2. 32 | We have the log-likelihood function: 33 | \[\ell_n(\lambda) = log(\prod_{i=1}^{n}e^{-\lambda} \frac{\lambda^{X_i}}{X_i!})=-n\lambda -\sum_{i=1}^{n}log(X_i!) + log(\lambda)\sum_{i=1}^{n}X_i \] 34 | Hence, 35 | \[\frac{\partial\ell_n(\lambda)}{\partial \lambda}= -n + \frac{1}{\lambda}\sum_{i=1}^{n}X_i \] 36 | Set this equals to 0, we get: $\lambda=\frac{1}{n}\sum_{i=1}^{n}X_i$.\\ 37 | We have: $\frac{\partial^2\ell_n(\lambda)}{\partial \lambda^2}= \frac{\partial (-1 + \frac{X}{\lambda})}{\partial \lambda}=\frac{-X}{\lambda^2}$. Hence, $I(\lambda)= -\mathbb{E}[\frac{\partial^2\ell_n(\lambda)}{\partial \lambda^2}]= -\mathbb{E}[-\frac{X}{\lambda^2}]=\frac{1}{\lambda}$\\ 38 | 39 | 40 | 41 | \item 3. 42 | We have the log-likelihood function: 43 | \[\ell_n(\lambda) = log(\lambda^n e^{-\lambda \sum_{i=1}^{n} X})=nlog(\lambda)-\lambda\sum_{i=1}^{n}X_i \] 44 | Hence, 45 | \[\frac{\partial\ell_n(\lambda)}{\partial \lambda}=\frac{n}{\lambda} - \sum_{i=1}^{n}X_i \] 46 | Set this equals to 0, we get: $\lambda=\frac{n}{\sum_{i=1}^{n}X_i}$.\\ 47 | We have: $\frac{\partial^2\ell_n(\lambda)}{\partial \lambda^2}= \frac{\partial (\frac{1}{\lambda} - X)}{\partial \lambda}=\frac{-1}{\lambda^2}$. Hence, $I(\lambda)= -\mathbb{E}[\frac{\partial^2\ell_n(\lambda)}{\partial \lambda^2}]= -\mathbb{E}[\frac{-1}{\lambda^2}]=\frac{1}{\lambda^2}$\\ 48 | \item 4. 49 | We have this result from problem 2 in Problem 3: \[\begin{cases} \mu=\bar{X_n}\\ \sigma^2=\frac{\sum_{i=1}^n(X_i-\bar{X_n})^2}{n}\end{cases}\] 50 | Recall from problem 2 in Problem 3: 51 | \[ \begin{cases} \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \mu} = \frac{1}{\sigma^2}(X - \mu) \\\frac{\partial \ell_n(\mu, \sigma^2)}{\partial \sigma^2} = \frac{-1}{2\sigma^2} + \frac{(x-\mu)^2}{2\sigma^4} \end{cases} \Leftrightarrow \begin{cases} \frac{\partial^2 \ell_n(\mu, \sigma^2)}{\partial \mu^2} = \frac{-1}{\sigma^2} \\ \frac{\partial^2 \ell_n(\mu, \sigma^2)}{\partial \mu \sigma^2}= \frac{\partial^2 \ell_n(\mu, \sigma^2)}{\partial \sigma^2 \mu}= \frac{-2(X-\mu)}{\sigma^3} \\ \frac{\partial^2 \ell_n(\mu, \sigma^2)}{\partial \sigma^2 \sigma^2} = \frac{1}{2\sigma^4} - \frac{(X-\mu)^2}{2\sigma^6} \end{cases}\] 52 | Hence, 53 | \[ I(\mu, \sigma^2)=\mathbb{E} \begin{bmatrix} 54 | \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \mu^2} & \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \mu \sigma^2}\\ 55 | \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \sigma^2 \mu} & \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \sigma^2 \sigma^2} 56 | \end{bmatrix} = -\mathbb{E} \begin{bmatrix} 57 | -\frac{1}{\sigma^2} & \frac{-2(X-\mu)}{\sigma^3} \\ 58 | \frac{-2(X-\mu)}{\sigma^3} & \frac{1}{2\sigma^4} - \frac{(X-\mu)^2}{2\sigma^6} 59 | \end{bmatrix}= \begin{bmatrix} 60 | \frac{1}{\sigma^2} & 0 \\ 61 | 0 & \frac{1}{2\sigma^4} 62 | \end{bmatrix} \] 63 | \item 5. 64 | We have the log-likelihood function: 65 | \[\ell_n(\lambda, a) = log(\prod_{i=1}^{n}\lambda e^{-\lambda(X_i-a)} \mathbb{1}_{x \geq a}) = \sum_{i=1}^{n}= nlog(\lambda) - \lambda \sum_{i=1}^{n}(X_i-a) +\sum_{i=1}^{n}log(\mathbb{1}_{X_i \geq a}) \] 66 | Hence, 67 | \[\frac{\partial\ell_n(\lambda, a)}{\partial a}= n\lambda \] 68 | Because $n\lambda $ for all $\lambda > 0$, $\ell_n(\lambda, a)$ is an increasing function of $a$ until $a > min(X_i)$. Hence $\ell_n(\lambda, a)$ is maximal with respect to $a$ when $a$ is made as large as possible without exceeding the minimum order statistic $\Longrightarrow a=min(X_i)$\\ 69 | Let's denote $y=x-a$, hence $f_{y}(y)=\frac{d}{dx}(x-a)f_{x}{y}=f_{x}(y)$ which is exponential distribution.\\ 70 | From question 3, we know $\lambda=\frac{n}{\sum_{i=1}^{n}Y_i}=\frac{n}{\sum_{i=1}^{n}(X_i-a)}$\\ 71 | We have: 72 | \[ \begin{cases} \frac{\partial\ell_n(\lambda, a)}{\partial \lambda} = \frac{1}{\lambda} - (X -a) \\ \frac{\partial\ell_n(\lambda, a)}{\partial a} = \lambda \end{cases} \Leftrightarrow \begin{cases} \frac{\partial^2\ell_n(\lambda, a)}{\partial \lambda^2} = \frac{-1}{\lambda^2} \\ \frac{\partial^2\ell_n(\lambda, a)}{\partial \lambda a} = \frac{\partial^2\ell_n(\lambda, a)}{\partial a\lambda}= 1 \\ \frac{\partial^2\ell_n(\lambda, a)}{\partial a^2} = 0 \end{cases}\] 73 | Hence, 74 | \[ I(\lambda , a)=\mathbb{E} \begin{bmatrix} 75 | \frac{\partial \ell_n(\lambda, a)}{\partial \lambda^2} & \frac{\partial \ell_n(\lambda, a)}{\partial \lambda a}\\ 76 | \frac{\partial \ell_n(\lambda, a)}{\partial a \lambda} & \frac{\partial \ell_n(\lambda, a)}{\partial a^2} 77 | \end{bmatrix} = -\mathbb{E} \begin{bmatrix} 78 | -\frac{1}{\lambda^2} & 1 \\ 79 | 1 & 0 80 | \end{bmatrix}= \begin{bmatrix} 81 | \frac{1}{\lambda^2} & 1 \\ 82 | 1 & 0 83 | \end{bmatrix} \] 84 | 85 | 86 | \item 6. 87 | We have the log-likelihood function: 88 | \[\ell_n(\mu, \sigma^2) = log(\prod_{i=1}^{n}\frac{1}{X_{i}\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2\sigma^2} (log(X_i) - \mu)^2})= -\frac{n}{2}log(2\pi) - \frac{n}{2}log(\sigma^2) -\sum_{i=1}^{n}log(X_{i}) -\frac{1}{2\sigma^2}\sum_{i=1}^{n}(log(X_i)-\mu)^2 \] 89 | Hence, 90 | \[\frac{\partial\ell_n(\mu, \sigma^2)}{\partial \mu}= \frac{1}{\sigma^2}\Big ( \sum_{i=1}^{n}log(X_i) -n\mu\Big )\] 91 | Set this equals to 0, we get: $\mu=\frac{1}{n}\sum_{i=1}^{n}log(X_i)$.\\ 92 | On the other hand, 93 | \[\frac{\partial\ell_n(\mu, \sigma^2)}{\partial \sigma^2} = \frac{-n}{2\sigma^2}+ \frac{\sum_{i=1}^{n} (log(X_i) - \mu)^2}{2\sigma^4} \] 94 | Set this equals to 0, we get: $\sigma^2=\frac{1}{n}\sum_{i=1}^{n}(log(X_i)-\mu)^2$.\\ 95 | 96 | 97 | We have: 98 | \[ \begin{cases} \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \mu} = \frac{1}{\sigma^2}(log(X) - \mu) \\\frac{\partial \ell_n(\mu, \sigma^2)}{\partial \sigma^2} = \frac{-1}{2\sigma^2} + \frac{(log(x)-\mu)^2}{2\sigma^4} \end{cases} \Leftrightarrow \begin{cases} \frac{\partial^2 \ell_n(\mu, \sigma^2)}{\partial \mu^2} = \frac{-1}{\sigma^2} \\ \frac{\partial^2 \ell_n(\mu, \sigma^2)}{\partial \mu \sigma^2}= \frac{\partial^2 \ell_n(\mu, \sigma^2)}{\partial \sigma^2 \mu}= \frac{-2(log(X)-\mu)}{\sigma^3} \\ \frac{\partial^2 \ell_n(\mu, \sigma^2)}{\partial \sigma^2 \sigma^2} = \frac{1}{2\sigma^4} - \frac{(log(X)-\mu)^2}{2\sigma^6} \end{cases}\] 99 | Hence, 100 | \[ I(\mu, \sigma^2)=\mathbb{E} \begin{bmatrix} 101 | \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \mu^2} & \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \mu \sigma^2}\\ 102 | \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \sigma^2 \mu} & \frac{\partial \ell_n(\mu, \sigma^2)}{\partial \sigma^2 \sigma^2} 103 | \end{bmatrix} = -\mathbb{E} \begin{bmatrix} 104 | -\frac{1}{\sigma^2} & \frac{-2(log(X)-\mu)}{\sigma^3} \\ 105 | \frac{-2(log(X)-\mu)}{\sigma^3} & \frac{1}{2\sigma^4} - \frac{(log(X)-\mu)^2}{2\sigma^6} 106 | \end{bmatrix}= \begin{bmatrix} 107 | \frac{1}{\sigma^2} & 0 \\ 108 | 0 & \frac{1}{2\sigma^4} 109 | \end{bmatrix} \] 110 | The last equation comes from the fact that $ln(X) \sim N(\mu, \sigma^2)$ because of the following reason:\\ 111 | Let $y=log(x)$ 112 | \[ f_{y}(y)=f_{x}(e^y)e^{y}=\frac{e^{y}}{e^{y}\sqrt{2\pi \sigma^2}} e^{-\frac{1}{2\sigma^2}(log(e^y)-\mu)^2}=\frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{1}{2\sigma^2}(y-\mu)^2} \] 113 | Hence, $y \sim N(\mu, \sigma^2) \Longrightarrow \mathbb{E}[y]=\mathbb{E}[log(x)]=\mu \Longrightarrow \mathbb{E}[\log(x) -\mu] = 0$. 114 | There is an another better approach for the first part of this question (maximum likelihood). Let $y=log(x)$, we have $y\sim N(\mu, \sigma^2)$ (which was proved a few lines above). Now, We come up with question 4.\\ 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | \end{problem} 128 | 129 | \begin{problem}{2} 130 | \item 1. We have the first moment $m_{1}=\mathbb{E}[X]=p$. Hence, $p_{MM}=\frac{1}{n}\sum_{i=1}^{n}X_i$.\\ 131 | \item 2. We have the first moment $m_{1}=\mathbb{E}[X]=\lambda$. Hence, $\lambda_{MM}=\frac{1}{n}\sum_{i=1}^{n}X_i$.\\ 132 | \item 3. We have the first moment $m_{1}=\mathbb{E}[X]=\frac{1}{\lambda}$. Hence, $\frac{1}{\lambda_{MM}}=\frac{1}{n}\sum_{i=1}^{n} \Leftrightarrow \lambda_{MM}=\frac{n}{\sum_{i=1}^{n}X_i}$.\\ 133 | \item 4. We have: 134 | \[ \begin{cases} m_{1}=\mathbb{E}[X]=\mu \\ m_{2}=\mathbb{E}[X^2]=\mu^2 + \sigma^2 \end{cases}\] 135 | Hence, 136 | \[ \begin{cases} \mu_{MM}=\frac{1}{n}\sum_{i=1}^{n}X_i \\ \mu_{MM}^2 + \sigma_{MM}^2 = \frac{1}{n}\sum_{i=1}^{n}X_{i}^{2} \end{cases} \Leftrightarrow \begin{cases} \mu_{MM}=\frac{1}{n}\sum_{i=1}^{n}X_i \\ \sigma_{MM}^2 = \frac{1}{n}\sum_{i=1}^{n}X_{i}^{2} - \mu_{MM}^2 \end{cases} \] 137 | \item 5. 138 | Let's denote $y=x-a$, from question 5 problem 1 we have $y \sim Exp(\lambda)$. Hence: 139 | \[ \begin{cases} \mathbb{E}[X]=\mathbb{E}[Y]+a=\frac{1}{\lambda} + a\\ 140 | \mathbb{E}[X^2]=\mathbb{E}[(Y+a)^2]= \frac{2}{\lambda^2} + \frac{2a}{\lambda} + a^2 \end{cases}\]. 141 | We have system equation: 142 | \[ \begin{cases} \frac{1}{\lambda_{MM}} + a_{MM}= \bar{X_n}\\ 143 | \frac{2}{\lambda_{MM}^2} + \frac{2a_{MM}}{\lambda_{MM}} + a_{MM}^2 = \bar{X_n}^2 \end{cases} \Leftrightarrow \begin{cases} \lambda_{MM} = \frac{1}{\sqrt{\bar{X_n}^2- \bar{X_n}}}\\ a = \bar{X_n} - \frac{1}{\lambda_{MM}} \end{cases}\]. 144 | 145 | \item 6. 146 | Let's denote $Y=log(X)$. From question 6 Problem 1 we know $Y \sim N(\mu, \sigma)$\\ 147 | Form question 4, we have 148 | \[ \begin{cases} \mu_{MM}=\frac{1}{n}\sum_{i=1}^{n}Y_i = \frac{1}{n}\sum_{i=1}^{n}log(X_i) \\ \sigma_{MM}^2 = \frac{1}{n}\sum_{i=1}^{n}Y_{i}^{2} - \mu_{MM}^2 = \frac{1}{n}\sum_{i=1}^{n}log(X_{i})^{2} - \mu_{MM}^2 \end{cases} \] 149 | 150 | 151 | 152 | \end{problem} 153 | \begin{problem}{3} 154 | \item 1. $\mathbb{P}[X=1]=\mathbb{P}[Exp(\lambda) > z]= e^{-\lambda z} $. Hence $X_{i}$'s follow Bernoulli distribution with parameter $p=e^{-\lambda z}$. 155 | \item 2. We have $\bar{X_n}=\frac{1}{n}\sum_{i=1}^{n}X_i$. By CLL, $\bar{X_n} \sim N(\mathbb{E}[X], \frac{\mathbb{V}[X]}{n})$. From question 1, we have $\mathbb{E}[X]=p=e^{-\lambda z}$ and $\mathbb{V}[X]=p(1-p)=e^{-\lambda z}(1- e^{-\lambda z})$. Hence, $\bar{X_n} \sim N(e^{-\lambda z}, \frac{e^{-\lambda z}(1- e^{-\lambda z})}{n} )$. 156 | \item 3. From question 1 problem 1 we have $\bar{X_n}$ is a consistent estimator of $p = e^{-\lambda z}$. Hence, $\frac{-log(\bar{X_n})}{z}$ is a consistent estimator of $\lambda$. Hence, $f(\bar{X_n})=\frac{-log(\bar{X_n})}{z}$ 157 | 158 | 159 | \item 4. By CLT, we have $\sqrt{n}(\bar{X_n} -p) \longrightarrow N(0, p(1-p))$\\ 160 | On the other hand, let $f(x)=\frac{-log(x)}{z} \Rightarrow \frac{\partial f(x)}{\partial x}=\frac{-1}{zx} \Rightarrow \frac{\partial f(x)}{\partial x} (p)=\frac{-1}{zp}$. 161 | Using Delta-method, we know: 162 | \[\sqrt{n}(f(\bar{X_n}) - f(p)) \longrightarrow N(0, (\frac{-1}{zp})^2 p(1-p)) \Leftrightarrow \sqrt{n}(f(\bar{X_n}) - \frac{-log(p)}{z}) \longrightarrow N(0, \frac{1-p}{z^2p})\] 163 | 164 | \item 5. 165 | Let denote $f(p)= \frac{1-p}{z^2p}$. Hence, $\frac{\partial f(p)}{\partial p}= \frac{-1}{z^2p^2} \leqslant 0 \Rightarrow argmin f(p)=1$. Hence, $e^{-\lambda z}=1 \Leftrightarrow g_{\lambda}(z)= ze^{-\lambda z}=z $ 166 | \item 6.\\ 167 | a) The statistical model is ($\mathbb{R}+$, ($Exp(\lambda))_{\lambda>0}$).\\ 168 | b) From question 3 problem 1, we know $I_{Y}(\lambda)=\frac{1}{\lambda^2}$.\\ 169 | c) From question 1 problem 1, we know $I_{X}(\lambda)=\frac{1}{p(1-p)}=\frac{1}{e^{-\lambda z}(1-e^{-\lambda z})}$.\\ 170 | d) I wonder what value of threshold z is? 171 | 172 | \end{problem} 173 | 174 | \end{document} -------------------------------------------------------------------------------- /PS5/ProblemSet5.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 4 | \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined 5 | \global\let\oldcontentsline\contentsline 6 | \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} 7 | \global\let\oldnewlabel\newlabel 8 | \gdef\newlabel#1#2{\newlabelxx{#1}#2} 9 | \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} 10 | \AtEndDocument{\ifx\hyper@anchor\@undefined 11 | \let\contentsline\oldcontentsline 12 | \let\newlabel\oldnewlabel 13 | \fi} 14 | \fi} 15 | \global\let\hyper@last\relax 16 | \gdef\HyperFirstAtBeginDocument#1{#1} 17 | \providecommand\HyField@AuxAddToFields[1]{} 18 | \providecommand\HyField@AuxAddToCoFields[2]{} 19 | -------------------------------------------------------------------------------- /PS5/ProblemSet5.log: -------------------------------------------------------------------------------- 1 | This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex 2019.9.8) 28 DEC 2019 09:50 2 | entering extended mode 3 | restricted \write18 enabled. 4 | %&-line parsing enabled. 5 | **ProblemSet5.tex 6 | (./ProblemSet5.tex 7 | LaTeX2e <2017-04-15> 8 | Babel <3.18> and hyphenation patterns for 84 language(s) loaded. 9 | (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls 10 | Document Class: article 2014/09/29 v1.4h Standard LaTeX document class 11 | (/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo 12 | File: size10.clo 2014/09/29 v1.4h Standard LaTeX file (size option) 13 | ) 14 | \c@part=\count79 15 | \c@section=\count80 16 | \c@subsection=\count81 17 | \c@subsubsection=\count82 18 | \c@paragraph=\count83 19 | \c@subparagraph=\count84 20 | \c@figure=\count85 21 | \c@table=\count86 22 | \abovecaptionskip=\skip41 23 | \belowcaptionskip=\skip42 24 | \bibindent=\dimen102 25 | ) 26 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty 27 | Package: hyperref 2018/02/06 v6.86b Hypertext links for LaTeX 28 | 29 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty 30 | Package: hobsub-hyperref 2016/05/16 v1.14 Bundle oberdiek, subset hyperref (HO) 31 | 32 | 33 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty 34 | Package: hobsub-generic 2016/05/16 v1.14 Bundle oberdiek, subset generic (HO) 35 | Package: hobsub 2016/05/16 v1.14 Construct package bundles (HO) 36 | Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) 37 | Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) 38 | Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) 39 | Package ifluatex Info: LuaTeX not detected. 40 | Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) 41 | Package ifvtex Info: VTeX not detected. 42 | Package: intcalc 2016/05/16 v1.2 Expandable calculations with integers (HO) 43 | Package: ifpdf 2017/03/15 v3.2 Provides the ifpdf switch 44 | Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) 45 | Package etexcmds Info: Could not find \expanded. 46 | (etexcmds) That can mean that you are not using pdfTeX 1.50 or 47 | (etexcmds) that some package has redefined \expanded. 48 | (etexcmds) In the latter case, load this package earlier. 49 | Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) 50 | Package: kvdefinekeys 2016/05/16 v1.4 Define keys (HO) 51 | Package: pdftexcmds 2018/01/21 v0.26 Utility functions of pdfTeX for LuaTeX (HO 52 | ) 53 | Package pdftexcmds Info: LuaTeX not detected. 54 | Package pdftexcmds Info: \pdf@primitive is available. 55 | Package pdftexcmds Info: \pdf@ifprimitive is available. 56 | Package pdftexcmds Info: \pdfdraftmode found. 57 | Package: pdfescape 2016/05/16 v1.14 Implements pdfTeX's escape features (HO) 58 | Package: bigintcalc 2016/05/16 v1.4 Expandable calculations on big integers (HO 59 | ) 60 | Package: bitset 2016/05/16 v1.2 Handle bit-vector datatype (HO) 61 | Package: uniquecounter 2016/05/16 v1.3 Provide unlimited unique counter (HO) 62 | ) 63 | Package hobsub Info: Skipping package `hobsub' (already loaded). 64 | Package: letltxmacro 2016/05/16 v1.5 Let assignment for LaTeX macros (HO) 65 | Package: hopatch 2016/05/16 v1.3 Wrapper for package hooks (HO) 66 | Package: xcolor-patch 2016/05/16 xcolor patch 67 | Package: atveryend 2016/05/16 v1.9 Hooks at the very end of document (HO) 68 | Package atveryend Info: \enddocument detected (standard20110627). 69 | Package: atbegshi 2016/06/09 v1.18 At begin shipout hook (HO) 70 | Package: refcount 2016/05/16 v3.5 Data extraction from label references (HO) 71 | Package: hycolor 2016/05/16 v1.8 Color options for hyperref/bookmark (HO) 72 | ) 73 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty 74 | Package: keyval 2014/10/28 v1.15 key=value parser (DPC) 75 | \KV@toks@=\toks14 76 | ) 77 | (/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty 78 | Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional 79 | ) 80 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty 81 | Package: auxhook 2016/05/16 v1.4 Hooks for auxiliary files (HO) 82 | ) 83 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty 84 | Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) 85 | ) 86 | \@linkdim=\dimen103 87 | \Hy@linkcounter=\count87 88 | \Hy@pagecounter=\count88 89 | 90 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def 91 | File: pd1enc.def 2018/02/06 v6.86b Hyperref: PDFDocEncoding definition (HO) 92 | ) 93 | \Hy@SavedSpaceFactor=\count89 94 | 95 | (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg 96 | File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive 97 | ) 98 | Package hyperref Info: Hyper figures OFF on input line 4509. 99 | Package hyperref Info: Link nesting OFF on input line 4514. 100 | Package hyperref Info: Hyper index ON on input line 4517. 101 | Package hyperref Info: Plain pages OFF on input line 4524. 102 | Package hyperref Info: Backreferencing OFF on input line 4529. 103 | Package hyperref Info: Implicit mode ON; LaTeX internals redefined. 104 | Package hyperref Info: Bookmarks ON on input line 4762. 105 | \c@Hy@tempcnt=\count90 106 | 107 | (/usr/share/texlive/texmf-dist/tex/latex/url/url.sty 108 | \Urlmuskip=\muskip10 109 | Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. 110 | ) 111 | LaTeX Info: Redefining \url on input line 5115. 112 | \XeTeXLinkMargin=\dimen104 113 | \Fld@menulength=\count91 114 | \Field@Width=\dimen105 115 | \Fld@charsize=\dimen106 116 | Package hyperref Info: Hyper figures OFF on input line 6369. 117 | Package hyperref Info: Link nesting OFF on input line 6374. 118 | Package hyperref Info: Hyper index ON on input line 6377. 119 | Package hyperref Info: backreferencing OFF on input line 6384. 120 | Package hyperref Info: Link coloring OFF on input line 6389. 121 | Package hyperref Info: Link coloring with OCG OFF on input line 6394. 122 | Package hyperref Info: PDF/A mode OFF on input line 6399. 123 | LaTeX Info: Redefining \ref on input line 6439. 124 | LaTeX Info: Redefining \pageref on input line 6443. 125 | \Hy@abspage=\count92 126 | \c@Item=\count93 127 | \c@Hfootnote=\count94 128 | ) 129 | Package hyperref Info: Driver (autodetected): hpdftex. 130 | 131 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def 132 | File: hpdftex.def 2018/02/06 v6.86b Hyperref driver for pdfTeX 133 | \Fld@listcount=\count95 134 | \c@bookmark@seq@number=\count96 135 | 136 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty 137 | Package: rerunfilecheck 2016/05/16 v1.8 Rerun checks for auxiliary files (HO) 138 | Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 139 | 82. 140 | ) 141 | \Hy@SectionHShift=\skip43 142 | ) 143 | (/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty 144 | Package: geometry 2010/09/12 v5.6 Page Geometry 145 | \Gm@cnth=\count97 146 | \Gm@cntv=\count98 147 | \c@Gm@tempcnt=\count99 148 | \Gm@bindingoffset=\dimen107 149 | \Gm@wd@mp=\dimen108 150 | \Gm@odd@mp=\dimen109 151 | \Gm@even@mp=\dimen110 152 | \Gm@layoutwidth=\dimen111 153 | \Gm@layoutheight=\dimen112 154 | \Gm@layouthoffset=\dimen113 155 | \Gm@layoutvoffset=\dimen114 156 | \Gm@dimlist=\toks15 157 | ) 158 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty 159 | Package: amsmath 2017/09/02 v2.17a AMS math features 160 | \@mathmargin=\skip44 161 | 162 | For additional information on amsmath, use the `?' option. 163 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty 164 | Package: amstext 2000/06/29 v2.01 AMS text 165 | 166 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty 167 | File: amsgen.sty 1999/11/30 v2.0 generic functions 168 | \@emptytoks=\toks16 169 | \ex@=\dimen115 170 | )) 171 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty 172 | Package: amsbsy 1999/11/29 v1.2d Bold Symbols 173 | \pmbraise@=\dimen116 174 | ) 175 | (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty 176 | Package: amsopn 2016/03/08 v2.02 operator names 177 | ) 178 | \inf@bad=\count100 179 | LaTeX Info: Redefining \frac on input line 213. 180 | \uproot@=\count101 181 | \leftroot@=\count102 182 | LaTeX Info: Redefining \overline on input line 375. 183 | \classnum@=\count103 184 | \DOTSCASE@=\count104 185 | LaTeX Info: Redefining \ldots on input line 472. 186 | LaTeX Info: Redefining \dots on input line 475. 187 | LaTeX Info: Redefining \cdots on input line 596. 188 | \Mathstrutbox@=\box26 189 | \strutbox@=\box27 190 | \big@size=\dimen117 191 | LaTeX Font Info: Redeclaring font encoding OML on input line 712. 192 | LaTeX Font Info: Redeclaring font encoding OMS on input line 713. 193 | \macc@depth=\count105 194 | \c@MaxMatrixCols=\count106 195 | \dotsspace@=\muskip11 196 | \c@parentequation=\count107 197 | \dspbrk@lvl=\count108 198 | \tag@help=\toks17 199 | \row@=\count109 200 | \column@=\count110 201 | \maxfields@=\count111 202 | \andhelp@=\toks18 203 | \eqnshift@=\dimen118 204 | \alignsep@=\dimen119 205 | \tagshift@=\dimen120 206 | \tagwidth@=\dimen121 207 | \totwidth@=\dimen122 208 | \lineht@=\dimen123 209 | \@envbody=\toks19 210 | \multlinegap=\skip45 211 | \multlinetaggap=\skip46 212 | \mathdisplay@stack=\toks20 213 | LaTeX Info: Redefining \[ on input line 2817. 214 | LaTeX Info: Redefining \] on input line 2818. 215 | ) 216 | (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty 217 | Package: amsthm 2017/10/31 v2.20.4 218 | \thm@style=\toks21 219 | \thm@bodyfont=\toks22 220 | \thm@headfont=\toks23 221 | \thm@notefont=\toks24 222 | \thm@headpunct=\toks25 223 | \thm@preskip=\skip47 224 | \thm@postskip=\skip48 225 | \thm@headsep=\skip49 226 | \dth@everypar=\toks26 227 | ) 228 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty 229 | Package: amssymb 2013/01/14 v3.01 AMS font symbols 230 | 231 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty 232 | Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support 233 | \symAMSa=\mathgroup4 234 | \symAMSb=\mathgroup5 235 | LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' 236 | (Font) U/euf/m/n --> U/euf/b/n on input line 106. 237 | )) 238 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty 239 | Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) 240 | 241 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty 242 | Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR) 243 | 244 | (/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty 245 | Package: trig 2016/01/03 v1.10 sin cos tan (DPC) 246 | ) 247 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg 248 | File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration 249 | ) 250 | Package graphics Info: Driver file: pdftex.def on input line 99. 251 | 252 | (/usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def 253 | File: pdftex.def 2018/01/08 v1.0l Graphics/color driver for pdftex 254 | )) 255 | \Gin@req@height=\dimen124 256 | \Gin@req@width=\dimen125 257 | ) 258 | (/usr/share/texlive/texmf-dist/tex/latex/tools/multicol.sty 259 | Package: multicol 2017/04/11 v1.8q multicolumn formatting (FMi) 260 | \c@tracingmulticols=\count112 261 | \mult@box=\box28 262 | \multicol@leftmargin=\dimen126 263 | \c@unbalance=\count113 264 | \c@collectmore=\count114 265 | \doublecol@number=\count115 266 | \multicoltolerance=\count116 267 | \multicolpretolerance=\count117 268 | \full@width=\dimen127 269 | \page@free=\dimen128 270 | \premulticols=\dimen129 271 | \postmulticols=\dimen130 272 | \multicolsep=\skip50 273 | \multicolbaselineskip=\skip51 274 | \partial@page=\box29 275 | \last@line=\box30 276 | \maxbalancingoverflow=\dimen131 277 | \mult@rightbox=\box31 278 | \mult@grightbox=\box32 279 | \mult@gfirstbox=\box33 280 | \mult@firstbox=\box34 281 | \@tempa=\box35 282 | \@tempa=\box36 283 | \@tempa=\box37 284 | \@tempa=\box38 285 | \@tempa=\box39 286 | \@tempa=\box40 287 | \@tempa=\box41 288 | \@tempa=\box42 289 | \@tempa=\box43 290 | \@tempa=\box44 291 | \@tempa=\box45 292 | \@tempa=\box46 293 | \@tempa=\box47 294 | \@tempa=\box48 295 | \@tempa=\box49 296 | \@tempa=\box50 297 | \@tempa=\box51 298 | \c@columnbadness=\count118 299 | \c@finalcolumnbadness=\count119 300 | \last@try=\dimen132 301 | \multicolovershoot=\dimen133 302 | \multicolundershoot=\dimen134 303 | \mult@nat@firstbox=\box52 304 | \colbreak@box=\box53 305 | \mc@col@check@num=\count120 306 | ) 307 | (/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty 308 | Package: array 2016/10/06 v2.4d Tabular extension package (FMi) 309 | \col@sep=\dimen135 310 | \extrarowheight=\dimen136 311 | \NC@list=\toks27 312 | \extratabsurround=\skip52 313 | \backup@length=\skip53 314 | ) (./ProblemSet5.aux) 315 | \openout1 = `ProblemSet5.aux'. 316 | 317 | LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 14. 318 | LaTeX Font Info: ... okay on input line 14. 319 | LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 14. 320 | LaTeX Font Info: ... okay on input line 14. 321 | LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 14. 322 | LaTeX Font Info: ... okay on input line 14. 323 | LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 14. 324 | LaTeX Font Info: ... okay on input line 14. 325 | LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 14. 326 | LaTeX Font Info: ... okay on input line 14. 327 | LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 14. 328 | LaTeX Font Info: ... okay on input line 14. 329 | LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 14. 330 | LaTeX Font Info: ... okay on input line 14. 331 | \AtBeginShipoutBox=\box54 332 | Package hyperref Info: Link coloring OFF on input line 14. 333 | 334 | (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty 335 | Package: nameref 2016/05/21 v2.44 Cross-referencing by name of section 336 | 337 | (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty 338 | Package: gettitlestring 2016/05/16 v1.5 Cleanup title references (HO) 339 | ) 340 | \c@section@level=\count121 341 | ) 342 | LaTeX Info: Redefining \ref on input line 14. 343 | LaTeX Info: Redefining \pageref on input line 14. 344 | LaTeX Info: Redefining \nameref on input line 14. 345 | 346 | (./ProblemSet5.out) (./ProblemSet5.out) 347 | \@outlinefile=\write3 348 | \openout3 = `ProblemSet5.out'. 349 | 350 | 351 | *geometry* driver: auto-detecting 352 | *geometry* detected driver: pdftex 353 | *geometry* verbose mode - [ preamble ] result: 354 | * driver: pdftex 355 | * paper: 356 | * layout: 357 | * layoutoffset:(h,v)=(0.0pt,0.0pt) 358 | * modes: 359 | * h-part:(L,W,R)=(72.26999pt, 469.75502pt, 72.26999pt) 360 | * v-part:(T,H,B)=(72.26999pt, 650.43001pt, 72.26999pt) 361 | * \paperwidth=614.295pt 362 | * \paperheight=794.96999pt 363 | * \textwidth=469.75502pt 364 | * \textheight=650.43001pt 365 | * \oddsidemargin=0.0pt 366 | * \evensidemargin=0.0pt 367 | * \topmargin=-37.0pt 368 | * \headheight=12.0pt 369 | * \headsep=25.0pt 370 | * \topskip=10.0pt 371 | * \footskip=30.0pt 372 | * \marginparwidth=65.0pt 373 | * \marginparsep=11.0pt 374 | * \columnsep=10.0pt 375 | * \skip\footins=9.0pt plus 4.0pt minus 2.0pt 376 | * \hoffset=0.0pt 377 | * \voffset=0.0pt 378 | * \mag=1000 379 | * \@twocolumnfalse 380 | * \@twosidefalse 381 | * \@mparswitchfalse 382 | * \@reversemarginfalse 383 | * (1in=72.27pt=25.4mm, 1cm=28.453pt) 384 | 385 | (/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii 386 | [Loading MPS to PDF converter (version 2006.09.02).] 387 | \scratchcounter=\count122 388 | \scratchdimen=\dimen137 389 | \scratchbox=\box55 390 | \nofMPsegments=\count123 391 | \nofMParguments=\count124 392 | \everyMPshowfont=\toks28 393 | \MPscratchCnt=\count125 394 | \MPscratchDim=\dimen138 395 | \MPnumerator=\count126 396 | \makeMPintoPDFobject=\count127 397 | \everyMPtoPDFconversion=\toks29 398 | ) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty 399 | Package: epstopdf-base 2016/05/15 v2.6 Base part for package epstopdf 400 | 401 | (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/grfext.sty 402 | Package: grfext 2016/05/16 v1.2 Manage graphics extensions (HO) 403 | ) 404 | Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 405 | 38. 406 | Package grfext Info: Graphics extension search list: 407 | (grfext) [.pdf,.png,.jpg,.mps,.jpeg,.jbig2,.jb2,.PDF,.PNG,.JPG,.JPE 408 | G,.JBIG2,.JB2,.eps] 409 | (grfext) \AppendGraphicsExtensions on input line 456. 410 | 411 | (/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg 412 | File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv 413 | e 414 | )) 415 | LaTeX Font Info: Try loading font information for U+msa on input line 18. 416 | 417 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd 418 | File: umsa.fd 2013/01/14 v3.01 AMS symbols A 419 | ) 420 | LaTeX Font Info: Try loading font information for U+msb on input line 18. 421 | 422 | (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd 423 | File: umsb.fd 2013/01/14 v3.01 AMS symbols B 424 | ) [1 425 | 426 | {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] 427 | Package atveryend Info: Empty hook `BeforeClearDocument' on input line 127. 428 | [2] 429 | Package atveryend Info: Empty hook `AfterLastShipout' on input line 127. 430 | (./ProblemSet5.aux) 431 | Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 127. 432 | Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 127. 433 | Package rerunfilecheck Info: File `ProblemSet5.out' has not changed. 434 | (rerunfilecheck) Checksum: D41D8CD98F00B204E9800998ECF8427E;0. 435 | Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 127. 436 | ) 437 | Here is how much of TeX's memory you used: 438 | 6545 strings out of 492982 439 | 93767 string characters out of 6134897 440 | 200176 words of memory out of 5000000 441 | 10050 multiletter control sequences out of 15000+600000 442 | 10591 words of font info for 42 fonts, out of 8000000 for 9000 443 | 1141 hyphenation exceptions out of 8191 444 | 28i,12n,32p,366b,385s stack positions out of 5000i,500n,10000p,200000b,80000s 445 | 460 | Output written on ProblemSet5.pdf (2 pages, 159435 bytes). 461 | PDF statistics: 462 | 82 PDF objects out of 1000 (max. 8388607) 463 | 61 compressed objects within 1 object stream 464 | 3 named destinations out of 1000 (max. 500000) 465 | 1 words of extra memory for PDF output out of 10000 (max. 10000000) 466 | 467 | -------------------------------------------------------------------------------- /PS5/ProblemSet5.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS5/ProblemSet5.pdf -------------------------------------------------------------------------------- /PS5/ProblemSet5.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS5/ProblemSet5.synctex.gz -------------------------------------------------------------------------------- /PS5/ProblemSet5.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt]{article} 2 | \usepackage{hyperref} 3 | \usepackage[margin=1in]{geometry} 4 | \usepackage{amsmath,amsthm,amssymb, graphicx, multicol, array} 5 | 6 | \newcommand{\N}{\mathbb{N}} 7 | \newcommand{\Z}{\mathbb{Z}} 8 | 9 | \newenvironment{problem}[2][Problem]{\begin{trivlist} 10 | \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} 11 | 12 | 13 | 14 | \begin{document} 15 | \title{Problem Set 5} 16 | \author{Hoang Nguyen, Huy Nguyen} 17 | \maketitle 18 | 19 | \begin{problem}{1} 20 | \item 1. 21 | From exercise 3 in Problem 2, If $Z \thicksim N(0,1)$, we get: $\mathbb{P}\Big[|\frac{\sqrt{n}(\bar{X_i}-\mu)}{\sigma}| \leqslant t\Big]=2\mathbb{P}[Z \leqslant t]-1$.\\ 22 | $X_1, X_2,...,X_n$ follow Poisson distribution. Hence, $\mathbb{E}[X_i]=\lambda$ and $\mathbb{V}[X_i]=\lambda$\\ 23 | From CTL: $\frac{\sqrt{n}(\bar{X_n}-\lambda)}{\sqrt{\lambda}} \thicksim N(0,1)$. Hence, $\mathbb{P}\Big [ |\frac{\sqrt{n}(\bar{X_n}-\lambda)}{\sqrt{\lambda}}| \leqslant t \Big]=2\mathbb{P}\Big [ \frac{\sqrt{n}(\bar{X_n}-\lambda)}{\sqrt{\lambda}} \leqslant t \Big ]-1=1-\alpha.$\\ 24 | Hence, $\mathbb{P}\Big [ \frac{\sqrt{n}(\bar{X_n}-\lambda)}{\sqrt{\lambda}} \leqslant t \Big ]=1- \frac{\alpha}{2}$. Therefore, $t=\phi^{-1}(1- \frac{\alpha}{2}).$\\ 25 | We have $ = \mathbb{P}\Big [ |\frac{\sqrt{n}(\bar{X_n}-\lambda)}{\sqrt{\lambda}}| \leqslant \phi^{-1}(1- \frac{\alpha}{2}) \Big ]\rightarrow (1-\alpha)$ as n tends to infinity. This equivalent to: 26 | \[ \mathbb{P}\Big[-\frac{\sqrt{n}(\bar{X_n}-\lambda)}{\sqrt{\lambda}} \leqslant \phi^{-1}(1- \frac{\alpha}{2}) \leqslant \frac{\sqrt{n}(\bar{X_n}-\lambda)}{\sqrt{\lambda}}\Big] \rightarrow 1-\alpha\] 27 | \[ \mathbb{P}\Big[\bar{X_n}- \frac{\phi^{-1}(1- \frac{\alpha}{2})\sqrt{\lambda}}{\sqrt{n}} \leqslant \lambda \leqslant \bar{X_n}+ \frac{\phi^{-1}(1- \frac{\alpha}{2})\sqrt{\lambda}}{\sqrt{n}} \Big] \rightarrow 1- \alpha\] 28 | We know: $\bar{X_n} \xrightarrow{P} \mathbb{E}[\bar{X_n}]=\lambda$\\ 29 | Hence, 30 | \[ \mathbb{P}\Big[\bar{X_n}- \frac{\phi^{-1}(1- \frac{\alpha}{2})\sqrt{\bar{X_n}}}{\sqrt{n}} \leqslant \lambda \leqslant \bar{X_n}+ \frac{\phi^{-1}(1- \frac{\alpha}{2})\sqrt{\bar{X_i}}}{\sqrt{n}} \Big] \geqslant 1- \alpha\] 31 | $\Rightarrow$ L=[$\bar{X_n}- \frac{\phi^{-1}(1- \frac{\alpha}{2})\sqrt{\bar{X_i}}}{\sqrt{n}}, \bar{X_n}+ \frac{\phi^{-1}(1- \frac{\alpha}{2})\sqrt{\bar{X_i}}}{\sqrt{n}}$] 32 | 33 | 34 | \item 2. 35 | From previous result, if $\lambda_0 \in L$, we do not reject $H_0$. Otherwise, there is evidence to reject $H_0$. 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | \end{problem} 54 | 55 | \begin{problem}{2} 56 | \item 1. 57 | Take derivative of Gaussian expression respect to $\mu_1, \sigma_1, \mu_2, \sigma_2$ and set it equals to 0, we get:\\ 58 | $\hat{\mu_1}=\bar{X_{n1}}$, $\hat{\mu_2}=\bar{Y_{n2}}$, $\hat{\sigma_1}^2=\frac{1}{n1}\sum_{i=1}^{n1} (X_i-\hat{\mu_1})^2$ and $\hat{\sigma_2}^2=\frac{1}{n1}\sum_{i=1}^{n2} (Y_i-\hat{\mu_2})^2$. 59 | 60 | \item 2. 61 | We have: $\frac{n_1\hat{\sigma_1}^2}{\sigma_1^2}=\frac{n_1\frac{1}{n1}\sum_{i=1}^{n_1} (X_i-\mu_1)^2}{\sigma_1^2}=\sum_{i=1}^{n_1}(\frac{X_i-\mu_1}{\sigma_1})^2$. Since $\frac{X_i-\mu_1}{\sigma_1}\sim N(0,1)$, hence $\frac{n_1\hat{\sigma_1}^2}{\sigma_1^2} \sim \chi_{n1}^2$.\\ 62 | By similar method, we get $\frac{n_2\hat{\sigma_2}^2}{\sigma_2^2} \sim \chi_{n2}^2$ 63 | 64 | 65 | 66 | 67 | 68 | 69 | \item 3. 70 | From previous result, $\frac{n_1\hat{\sigma_1}^2}{\sigma_1^2} + \frac{n_2\hat{\sigma_2}^2}{\sigma_2^2} \sim \chi_{n1}^2 + \chi_{n2}^2 \sim \chi_{n1+n2}^2$. 71 | 72 | 73 | \item 4. 74 | From CTT, $\bar{X_{n1}} \sim N(\mu_1, \sigma_1^2)$, $\bar{X_{n2}} \sim N(\mu_2, \sigma_2^2)$. Hence, $\Delta=\hat{\mu_1}-\hat{\mu_2}=\bar{X_{n1}}- \bar{X_{n2}} \sim N(\mu_1-\mu_2, \sigma_1^2+\sigma_2^2)$. 75 | 76 | 77 | \item 5. 78 | From the previous question, we get $\Delta=\hat{\mu_1}-\hat{\mu_2}=\bar{X_{n1}}- \bar{X_{n2}} \sim N(\mu_1-\mu_2, \sigma_1^2+\sigma_2^2)$.\\ 79 | Under $H_0$: $\Delta \sim N(0,\sqrt{ \sigma_1^2+\sigma_2^2})$. Hence, $\frac{\Delta}{\sigma_1^2+\sigma_2^2} \rightarrow \frac{\Delta}{\sqrt{\hat{\sigma_1}^2 + \hat{\sigma_2}^2}} \sim N(0,1)$.\\ 80 | Let's denote test statistic $T=\Big | \frac{\Delta}{\sqrt{\hat{\sigma_1}^2 + \hat{\sigma_2}^2}}\Big |$. We now $\mathbb{P}[T > c]=\alpha$, hence $c=1-\phi^{-1}(\frac{\alpha}{2})$.\\ 81 | Conclusion: If $T>c$, we reject $H_0$, otherwise, we fail to reject $H_0$. 82 | 83 | 84 | \item 6. 85 | In this case, from the previous qeuestion, we have $T=\Big | \frac{\Delta}{\sqrt{\hat{\sigma_1}^2 + \hat{\sigma_2}^2}}\Big |=\frac{8.43-8.07}{\sqrt{0.22+0.17}}=0.5765$. Morever, $c=\phi^{-1}(1-\frac{\alpha}{2})=\phi^{-1}(0.975)=1.96$. Because $T0.5765]=1-\mathbb{P}[|N(0,1)| \leqslant 0.5765]=1-(2\mathbb{P}[N(0,1) \leqslant 0.5765]-1)=2-2\phi(0.5765)=0.5642$. 87 | 88 | 89 | 90 | 91 | \end{problem} 92 | 93 | \begin{problem}{3} 94 | Let denote the function $\mathbb{R}^{2} \longrightarrow \mathbb{R}: g(\mu, \sigma^2)=\mu -\sqrt{\sigma^2}$. We have: 95 | \[ \frac{\partial g}{\partial \mu}=1 \] 96 | \[ \frac{\partial}{\partial \sigma^{2}}= -\frac{1}{2\sigma}\] 97 | Let denote $\hat{\mu}=\bar{X_n}$ and $\hat{\sigma}^{2}=\frac{1}{n}\sum(X_i-\hat{\mu})^{2}$. Hence, by \href{https://alecospapadopoulos.files.wordpress.com/2016/03/asympt-distr-of-sample-variance-dichotomous-rvs-technical-report.pdf}{this result}: 98 | \[ \sqrt{n}(\hat{\mu}-\mu) \longrightarrow N(0, \sigma^2).\] 99 | \[ \sqrt{n}(\hat{\sigma^2}- \sigma^2) \longrightarrow N(0, 2\sigma^{4}) \] 100 | Hence $(\hat{\mu}, \hat{\sigma^2})$ is asymptotically normal with covariance matrix: 101 | \[\sum= 102 | \begin{bmatrix} 103 | \sigma^2 & 0\\ 104 | 0 & 2\sigma^4 105 | \end{bmatrix} 106 | \] 107 | Using Delta method we have: 108 | \[ \sqrt{n}(g(\hat{\mu}, \hat{\sigma}^2) - g(\mu, \sigma^2)) \longrightarrow N(0, \nabla_{\mu, \sigma^2}^{T} \Sigma \nabla_{\mu, \sigma^2} ))\] 109 | Where $\nabla_{\mu, \sigma^2}= \nabla g(\mu, \sigma^2)=[1 -\frac{1}{2\sigma}]^{T}$. Hence $\nabla_{\mu, \sigma^2}^{T} \Sigma \nabla_{\mu, \sigma^2}=[1 -\frac{1}{2\sigma}]\begin{bmatrix} 110 | \sigma^2 & 0\\ 111 | 0 & 2\sigma^4 112 | \end{bmatrix} 113 | \begin{bmatrix} 114 | 1 \\ 115 | -\frac{1}{2\sigma} 116 | \end{bmatrix}=\frac{3}{2}\sigma^2$.\\ 117 | Hence $ \sqrt{n}(g(\hat{\mu}, \hat{\sigma}^2) - g(\mu, \sigma^2)) \longrightarrow N(0, \frac{3}{2}\sigma^2 )) $ which equivalents to $ \sqrt{n}\frac{\sqrt{\frac{2}{3}}}{\sigma}(g(\hat{\mu}, \hat{\sigma}^2) - g(\mu, \sigma^2)) \longrightarrow N(0, 1)$. Computing second order Delta method, we have: 118 | \[ n g(\hat{\mu}, \hat{\sigma}^2)^{T} \frac{2}{3\sigma^2} g(\hat{\mu}, \hat{\sigma}^2) \longrightarrow \chi_{1}^{2}\] 119 | Let denote c is $(1-\alpha)$-quantile of Kai-square with 1 degree of fredom. Because this is one-sided test statistic, if $T>c$ we reject null hypothesis, othewise, we fail to reject null hypothesis.\\ 120 | Consider null hypothesis $H_0:$ $\mu = \sigma$\\ 121 | In our case, under null hypothesis, $T=n g(\hat{\mu}, \hat{\sigma}^2)^{T} \frac{2}{3\sigma^2} g(\hat{\mu}, \hat{\sigma}^2) = 100 \times (2.41-\sqrt{5.2}) \times \frac{2}{3\times 5.2} \times (2.41-\sqrt{5.2})= 0.215 $ and $c=3.841$. Since $T we fail to reject null hypothesis\\ 123 | At level 0.1, we know $c=2.706$. Since $Tc]=\alpha$, hence $c=1-\phi^{-1}(\frac{\alpha}{2})$\\ 26 | Conclusion: If $T>c$, we reject $H_0$. Otherwise, we fail to reject $H_0$. 27 | 28 | 29 | \item 2. 30 | Let denote $\hat{\lambda}=argmax_{\theta \in \{0; +\infty\}} l(\theta)$ and $\hat{\lambda^c}=argmax_{\theta \in H_0} l(\theta)$ where $l(\theta)$ is likehood funtion of $\theta$. Using Likelood ratio test, we have test stattistic: $T=2\Big(l(\hat{\lambda})-l(\hat{\lambda^c})\Big)$. \\ 31 | The difference in the number of parameters for the two models is 1 because we only consider paramer $\lambda \in \mathbb{R}$. Hence $T \sim \chi_{1}^{2}$. With asymptotic level test $\alpha$, $\mathbb{P}[T>c]=\alpha$, hence c is the $(1-\alpha)$-quantile of $\chi^{2}_{1}$. 32 | Conclusion: If $T>c$, we reject $H_0$. Otherwise, we fail to reject $H_0$. 33 | 34 | \item 3. 35 | a) Because $X_1, X_2,...X_n$ follow exponential, hence $\l(\lambda)=nln(\lambda)- \lambda \sum_{i=1}^{n}X_i$. Because there are 50 consecutive calls and these observations have average value is 0.98, hence $ \sum_{i=1}^{n}X_i=0.98x50=49$.\\ 36 | Now we have log-likelihood function $l(\lambda)=50ln(\lambda)- 49\lambda$. Taking derivate respect to $\lambda$, we get $l'(\lambda)= \frac{50}{\lambda}-49$. Setting this equals to zeros we get $\hat{\lambda}=\frac{50}{49}$, hence $l(\hat{\lambda})=-48.989$\\ 37 | Because $l'(\lambda) > 0$ for all $\theta$ under $H_0$, hence $\hat{\lambda^c}=1 => l(\hat{\lambda^c})=-49$\\ 38 | Based on previous questions, we have $T=2\Big(l(\hat{\lambda})-l(\hat{\lambda^c})\Big)= 2(-48.989+49)=0.022$. With asymptotic level test $\alpha=0.05$, we get $c=3.814$\\ 39 | Because $T=0.022<3.814$, hence we fail to reject null hypothesis.\\ 40 | b) $p-value=\mathbb{P}[\chi_{1}^{2}>0.022]=0.8829$ 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | \end{problem} 62 | 63 | \begin{problem}{2} 64 | \item 1. 65 | Take derivative of Gaussian expression respect to $\mu_1, \sigma_1,$ and set it equals to 0, we get:\\ 66 | $\hat{\mu}=\bar{X_{n}}$ and $\hat{\sigma}^2=\frac{1}{n}\sum_{i=1}^{n} (X_i-\hat(\mu))^2$. 67 | 68 | \item 2. 69 | Cochran’s theorem implies that for $X_1, X_2,...X_n$ follow Gaussian distribution with $\mu$ and $\sigma^2$, we have: 70 | \[\sqrt{n-1}\frac{\bar{X_n}-\mu}{\sqrt{S_n}} \sim t_{n-1} \] 71 | From previous question, we have $\hat{\mu}=\bar{X_{n}}$ and $S_n=\hat{\sigma}^2=\frac{1}{n}\sum_{i=1}^{n} (X_i-\mu)^2$. Plugging this to the equation above, we get: 72 | \[S=\sqrt{n-1}\frac{\hat{\mu}- \mu}{\sqrt{\hat{\sigma}^2}} \sim t_{n-1} \] 73 | 74 | \item 3 75 | From previous question, we know $S \sim t_{n-1}$, so we will bulid Student test based on this intuition with test statistic S.\\ 76 | Let denote $c$ is the ($1-\alpha$)-quantile of Student distribution with $n-1$ degrees of freedom. That means $\mathbb{P}[t_{n-1}> c]=\alpha$.\\ 77 | Based on the previous question, we have test statistic: $T=\sqrt{n-1}\frac{\hat{\mu}}{\sqrt{\hat{\sigma}^2}}= \sqrt{n-1}\frac{\bar{X_n}}{\sqrt{\frac{1}{n}\sum_{i=1}^{n} (X_i-\bar{X_n})^2}} \sim t_{n-1}$. Concretely, the graph below show the t-test diagram with Student test (n-1 dgrees of freedom) and level test $\alpha=0.05$, for example. 78 | 79 | \includegraphics[totalheight=5cm]{img.jpg}\\ 80 | Conclusion: If $T>c$ that means $H_0$ fall into reject region so we reject $H_0$,. Otherwise, we are fail to reject $H_0$ 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | \end{problem} 96 | 97 | \begin{problem}{3} 98 | \item 1.\\ 99 | $\bullet$ Prove X and Y are independent implies $r=pq$:\\ 100 | From Bayes rule, we have $\mathbb{P}[X=1|Y=1]=\frac{\mathbb{P}[X=1, Y=1]}{\mathbb{P}[Y=1]}$ (1) \\ 101 | From X and Y are independent, we have $\mathbb{P}[X=1|Y=1]=\mathbb{P}[X=1] $. Plugging this to (1), we have: 102 | $\mathbb{P}[X=1] \mathbb{P}[Y=1]=\mathbb{P}[X=1, Y=1] \Leftrightarrow r=pq$.\\ 103 | $\bullet$ Prove $r=pq$ implies X and Y are independent: \\ 104 | By defination of independent event, we have if $\mathbb{P}[X=1]\mathbb{P}[Y=1]=\mathbb{P}[X=1, Y=1] \Leftrightarrow r=pq$, X and Y are independent.\\ 105 | Hence X and Y are independent iff $r=pq$. 106 | \item 2.\\ 107 | a)\\ 108 | $\bullet$ By Markov's inequality: 109 | \[ \mathbb{P}\Big[|\hat{p} - p| > \epsilon \Big] \leqslant \mathbb{P}\Big[\hat{p} - p > \epsilon \Big] \leqslant \frac{\mathbb{E}[\hat{p}-p]}{\epsilon} = \frac{\mathbb{E}[\hat{p}]- p}{\epsilon}=0 \] 110 | Hence, $\hat{p}$ is consistent estimators of $p$.\\ 111 | Using similar method, we get $\hat{q}$ and $\hat{r}$ are consistent estimator of q and r, respectively.\\ 112 | b)\\ 113 | From CTT, we have $\sqrt{n}\frac{\bar{X_n}-p}{\sqrt{p(1-p)}} \sim N(0,1) $. Hence, $\sqrt{n}(\hat{p}-p) \sim N(0, p(1-p))$. Using similar method, we get $\sqrt{n}(\hat{q}-q) \sim N(0, q(1-q))$ and $\sqrt{n}(\hat{r}-r) \sim N(0, r(1-r))$.\\ 114 | Hence, $(\hat{p}, \hat{q}, \hat{r})$ is asymptotically normal with covariance matrix 115 | \[ \Sigma 116 | = 117 | \begin{bmatrix} 118 | p(1-p) & 0 & 0\\ 119 | 0 & q(1-q) & 0\\ 120 | 0 & 0 & r(1-r) 121 | \end{bmatrix} 122 | \] 123 | c)\\ 124 | From previous question we have: $\sqrt{n}(\hat{\theta} -\theta ) \sim N(0, \Sigma)$ where $\theta=\begin{bmatrix} 125 | r & p & q 126 | 127 | \end{bmatrix}^{T} $ and $\Sigma=\begin{bmatrix} 128 | r(1-r) & 0 & 0\\ 129 | 0 & p(1-p) & 0\\ 130 | 0 & 0 & q(1-q) 131 | 132 | \end{bmatrix} $\\ 133 | Let's denote $\theta= \begin{bmatrix} 134 | r & p & q 135 | 136 | \end{bmatrix}^{T} $, $\hat{\theta}= \begin{bmatrix} 137 | \hat{r} & \hat{p} & \hat{q} 138 | 139 | \end{bmatrix}^{T} $ and function $\mathbb{R}^3 \rightarrow \mathbb{R}: f(\theta)=\theta_1-\theta_2\theta_3$.\\ 140 | We have $\sqrt{n}(\hat{r}-\hat{p}\hat{q})=\sqrt{n}(f(\hat{\theta})-f(\theta))$ where $\theta=\begin{bmatrix} 141 | r & p & q 142 | 143 | \end{bmatrix}^{T} $. Using Delta-method we have: 144 | \[ \sqrt{n}(f(\hat{\theta})-f(\theta)) \sim N(0, \nabla_{\theta}^{T} \Sigma \nabla_{\theta} ) \] 145 | Where $\nabla_{\theta}=\frac{\partial f}{\partial \theta}(\theta)$ and $\Sigma$ is the covariance matrix of asymptotically normal distributiion of $\theta$. With $\theta=\begin{bmatrix} 146 | r & p & q 147 | 148 | \end{bmatrix}^{T} $ we have: \[ \nabla_{\theta}= \frac{\partial f}{\partial \theta}(\theta)= 149 | \theta=\begin{bmatrix} 150 | 1 & -q & -p 151 | 152 | \end{bmatrix}^{T} 153 | \] 154 | Hence $V=\nabla_{\theta}^{T} \Sigma \nabla_{\theta}=\begin{bmatrix} 155 | 1 & -q & -p 156 | 157 | \end{bmatrix} \begin{bmatrix} 158 | r(1-r) & 0 & 0\\ 159 | 0 & p(1-p) & 0\\ 160 | 0 & 0 & q(1-q) 161 | 162 | \end{bmatrix} \begin{bmatrix} 163 | 1 \\ 164 | -q\\ 165 | -p\\ 166 | \end{bmatrix}=$.\\ 167 | 168 | Let X, Y be two Bernoulli random variables and denote by $p=\mathbb{P}[X=1]$, $q=\mathbb{P}[Y=1]$ and $r=\mathbb{P}[X=1, Y=1]$.\\ 169 | 1. Prove that X and Y are independent if and only if $r=pq$. 170 | 2. Let $(X_1, Y_1),...,(X_n, Y_n)$ be a sample of n i.i.d copies of $(X,Y)$. Based on this sample, we want to test whether X and Y are independent, i.e, where ther $r=pq$.\\ 171 | a) Define $\hat{p}=\frac{1}{n}\sum_{i=1}^{n}X_i$, $\hat{q}=\frac{1}{n}\sum_{i=1}^{n}Y_i$ and $\hat{r}=\frac{1}{n}\sum_{i=1}^{n}X_{i}Y_{i}$. Prrove taht these test are, respectively, consistent estimators of p, q and r.\\ 172 | b) Show that the vector $(\hat{p}, \hat{q}, \hat{r})$ is asymptotically normal and find the asymptotic covariance matrix.\\ 173 | c) Using the previous question combined with the Delta-method, prove that: 174 | \[\sqrt{n}(\hat{r}- \hat{p}\hat{q}-(r-pq)) \longrightarrow N(0, V) \] 175 | where V depends on p,q and r.\\ 176 | d) Consider the following hypotheses:\\ 177 | $H_0$: "X and Y are independent" vs. $H_1$: "X and Y are not independent".\\ 178 | Assuming that $H_0$ is true, show that $V=pq(1-p)(1-q)$ and propose a consistent estimator of V.\\ 179 | e) Using last two questions, propose a test with asymptotic $\alpha$, for any $\alpha \in (0,1)$. 180 | 181 | 182 | \end{problem} 183 | 184 | \end{document} -------------------------------------------------------------------------------- /PS6/img.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS6/img.jpg -------------------------------------------------------------------------------- /PS7/ProlemSet7_solustion.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | -------------------------------------------------------------------------------- /PS7/ProlemSet7_solustion.log: -------------------------------------------------------------------------------- 1 | This is pdfTeX, Version 3.14159265-2.6-1.40.20 (MiKTeX 2.9.7140 64-bit) (preloaded format=pdflatex 2019.9.10) 8 JAN 2020 09:31 2 | entering extended mode 3 | **./ProlemSet7_solustion.tex 4 | (ProlemSet7_solustion.tex 5 | LaTeX2e <2018-12-01> 6 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/base\article.cls" 7 | Document Class: article 2018/09/03 v1.4i Standard LaTeX document class 8 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/base\size10.clo" 9 | File: size10.clo 2018/09/03 v1.4i Standard LaTeX file (size option) 10 | ) 11 | \c@part=\count80 12 | \c@section=\count81 13 | \c@subsection=\count82 14 | \c@subsubsection=\count83 15 | \c@paragraph=\count84 16 | \c@subparagraph=\count85 17 | \c@figure=\count86 18 | \c@table=\count87 19 | \abovecaptionskip=\skip41 20 | \belowcaptionskip=\skip42 21 | \bibindent=\dimen102 22 | ) 23 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/geometry\geometry.s 24 | ty" 25 | Package: geometry 2018/04/16 v5.8 Page Geometry 26 | 27 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics\keyval.sty 28 | " 29 | Package: keyval 2014/10/28 v1.15 key=value parser (DPC) 30 | \KV@toks@=\toks14 31 | ) 32 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\ifpdf.st 33 | y" 34 | Package: ifpdf 2018/09/07 v3.3 Provides the ifpdf switch 35 | ) 36 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\ifvtex.s 37 | ty" 38 | Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) 39 | Package ifvtex Info: VTeX not detected. 40 | ) 41 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/ifxetex\ifxetex.s 42 | ty" 43 | Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional 44 | ) 45 | \Gm@cnth=\count88 46 | \Gm@cntv=\count89 47 | \c@Gm@tempcnt=\count90 48 | \Gm@bindingoffset=\dimen103 49 | \Gm@wd@mp=\dimen104 50 | \Gm@odd@mp=\dimen105 51 | \Gm@even@mp=\dimen106 52 | \Gm@layoutwidth=\dimen107 53 | \Gm@layoutheight=\dimen108 54 | \Gm@layouthoffset=\dimen109 55 | \Gm@layoutvoffset=\dimen110 56 | \Gm@dimlist=\toks15 57 | 58 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/geometry\geometry.c 59 | fg")) ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/bbm\bbm.sty" 60 | Package: bbm 1999/03/15 V 1.2 provides fonts for set symbols - TH 61 | LaTeX Font Info: Overwriting math alphabet `\mathbbm' in version `bold' 62 | (Font) U/bbm/m/n --> U/bbm/bx/n on input line 33. 63 | LaTeX Font Info: Overwriting math alphabet `\mathbbmss' in version `bold' 64 | (Font) U/bbmss/m/n --> U/bbmss/bx/n on input line 35. 65 | ) ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amsmath.s 66 | ty" 67 | Package: amsmath 2018/12/01 v2.17b AMS math features 68 | \@mathmargin=\skip43 69 | 70 | For additional information on amsmath, use the `?' option. 71 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amstext.sty 72 | " 73 | Package: amstext 2000/06/29 v2.01 AMS text 74 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amsgen.sty 75 | " 76 | File: amsgen.sty 1999/11/30 v2.0 generic functions 77 | \@emptytoks=\toks16 78 | \ex@=\dimen111 79 | )) 80 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amsbsy.sty" 81 | Package: amsbsy 1999/11/29 v1.2d Bold Symbols 82 | \pmbraise@=\dimen112 83 | ) ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsmath\amsopn.st 84 | y" 85 | Package: amsopn 2016/03/08 v2.02 operator names 86 | ) 87 | \inf@bad=\count91 88 | LaTeX Info: Redefining \frac on input line 223. 89 | \uproot@=\count92 90 | \leftroot@=\count93 91 | LaTeX Info: Redefining \overline on input line 385. 92 | \classnum@=\count94 93 | \DOTSCASE@=\count95 94 | LaTeX Info: Redefining \ldots on input line 482. 95 | LaTeX Info: Redefining \dots on input line 485. 96 | LaTeX Info: Redefining \cdots on input line 606. 97 | \Mathstrutbox@=\box27 98 | \strutbox@=\box28 99 | \big@size=\dimen113 100 | LaTeX Font Info: Redeclaring font encoding OML on input line 729. 101 | LaTeX Font Info: Redeclaring font encoding OMS on input line 730. 102 | \macc@depth=\count96 103 | \c@MaxMatrixCols=\count97 104 | \dotsspace@=\muskip10 105 | \c@parentequation=\count98 106 | \dspbrk@lvl=\count99 107 | \tag@help=\toks17 108 | \row@=\count100 109 | \column@=\count101 110 | \maxfields@=\count102 111 | \andhelp@=\toks18 112 | \eqnshift@=\dimen114 113 | \alignsep@=\dimen115 114 | \tagshift@=\dimen116 115 | \tagwidth@=\dimen117 116 | \totwidth@=\dimen118 117 | \lineht@=\dimen119 118 | \@envbody=\toks19 119 | \multlinegap=\skip44 120 | \multlinetaggap=\skip45 121 | \mathdisplay@stack=\toks20 122 | LaTeX Info: Redefining \[ on input line 2844. 123 | LaTeX Info: Redefining \] on input line 2845. 124 | ) 125 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amscls\amsthm.sty" 126 | Package: amsthm 2017/10/31 v2.20.4 127 | \thm@style=\toks21 128 | \thm@bodyfont=\toks22 129 | \thm@headfont=\toks23 130 | \thm@notefont=\toks24 131 | \thm@headpunct=\toks25 132 | \thm@preskip=\skip46 133 | \thm@postskip=\skip47 134 | \thm@headsep=\skip48 135 | \dth@everypar=\toks26 136 | ) 137 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsfonts\amssymb.st 138 | y" 139 | Package: amssymb 2013/01/14 v3.01 AMS font symbols 140 | 141 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsfonts\amsfonts.s 142 | ty" 143 | Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support 144 | \symAMSa=\mathgroup4 145 | \symAMSb=\mathgroup5 146 | LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' 147 | (Font) U/euf/m/n --> U/euf/b/n on input line 106. 148 | )) 149 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics\graphicx.s 150 | ty" 151 | Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) 152 | 153 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics\graphics.s 154 | ty" 155 | Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR) 156 | 157 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics\trig.sty" 158 | Package: trig 2016/01/03 v1.10 sin cos tan (DPC) 159 | ) 160 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics-cfg\graphi 161 | cs.cfg" 162 | File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration 163 | ) 164 | Package graphics Info: Driver file: pdftex.def on input line 99. 165 | 166 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/graphics-def\pdftex 167 | .def" 168 | File: pdftex.def 2018/01/08 v1.0l Graphics/color driver for pdftex 169 | )) 170 | \Gin@req@height=\dimen120 171 | \Gin@req@width=\dimen121 172 | ) 173 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/tools\multicol.sty" 174 | Package: multicol 2018/12/27 v1.8v multicolumn formatting (FMi) 175 | \c@tracingmulticols=\count103 176 | \mult@box=\box29 177 | \multicol@leftmargin=\dimen122 178 | \c@unbalance=\count104 179 | \c@collectmore=\count105 180 | \doublecol@number=\count106 181 | \multicoltolerance=\count107 182 | \multicolpretolerance=\count108 183 | \full@width=\dimen123 184 | \page@free=\dimen124 185 | \premulticols=\dimen125 186 | \postmulticols=\dimen126 187 | \multicolsep=\skip49 188 | \multicolbaselineskip=\skip50 189 | \partial@page=\box30 190 | \last@line=\box31 191 | \maxbalancingoverflow=\dimen127 192 | \mult@rightbox=\box32 193 | \mult@grightbox=\box33 194 | \mult@gfirstbox=\box34 195 | \mult@firstbox=\box35 196 | \@tempa=\box36 197 | \@tempa=\box37 198 | \@tempa=\box38 199 | \@tempa=\box39 200 | \@tempa=\box40 201 | \@tempa=\box41 202 | \@tempa=\box42 203 | \@tempa=\box43 204 | \@tempa=\box44 205 | \@tempa=\box45 206 | \@tempa=\box46 207 | \@tempa=\box47 208 | \@tempa=\box48 209 | \@tempa=\box49 210 | \@tempa=\box50 211 | \@tempa=\box51 212 | \@tempa=\box52 213 | \c@columnbadness=\count109 214 | \c@finalcolumnbadness=\count110 215 | \last@try=\dimen128 216 | \multicolovershoot=\dimen129 217 | \multicolundershoot=\dimen130 218 | \mult@nat@firstbox=\box53 219 | \colbreak@box=\box54 220 | \mc@col@check@num=\count111 221 | ) ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/tools\array.sty" 222 | Package: array 2018/12/30 v2.4k Tabular extension package (FMi) 223 | \col@sep=\dimen131 224 | \ar@mcellbox=\box55 225 | \extrarowheight=\dimen132 226 | \NC@list=\toks27 227 | \extratabsurround=\skip51 228 | \backup@length=\skip52 229 | \ar@cellbox=\box56 230 | ) (ProlemSet7_solustion.aux) 231 | \openout1 = `ProlemSet7_solustion.aux'. 232 | 233 | LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 15. 234 | LaTeX Font Info: ... okay on input line 15. 235 | LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 15. 236 | LaTeX Font Info: ... okay on input line 15. 237 | LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 15. 238 | LaTeX Font Info: ... okay on input line 15. 239 | LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 15. 240 | LaTeX Font Info: ... okay on input line 15. 241 | LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 15. 242 | LaTeX Font Info: ... okay on input line 15. 243 | LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 15. 244 | LaTeX Font Info: ... okay on input line 15. 245 | 246 | *geometry* driver: auto-detecting 247 | *geometry* detected driver: pdftex 248 | *geometry* verbose mode - [ preamble ] result: 249 | * driver: pdftex 250 | * paper: 251 | * layout: 252 | * layoutoffset:(h,v)=(0.0pt,0.0pt) 253 | * modes: 254 | * h-part:(L,W,R)=(72.26999pt, 469.75502pt, 72.26999pt) 255 | * v-part:(T,H,B)=(72.26999pt, 650.43001pt, 72.26999pt) 256 | * \paperwidth=614.295pt 257 | * \paperheight=794.96999pt 258 | * \textwidth=469.75502pt 259 | * \textheight=650.43001pt 260 | * \oddsidemargin=0.0pt 261 | * \evensidemargin=0.0pt 262 | * \topmargin=-37.0pt 263 | * \headheight=12.0pt 264 | * \headsep=25.0pt 265 | * \topskip=10.0pt 266 | * \footskip=30.0pt 267 | * \marginparwidth=65.0pt 268 | * \marginparsep=11.0pt 269 | * \columnsep=10.0pt 270 | * \skip\footins=9.0pt plus 4.0pt minus 2.0pt 271 | * \hoffset=0.0pt 272 | * \voffset=0.0pt 273 | * \mag=1000 274 | * \@twocolumnfalse 275 | * \@twosidefalse 276 | * \@mparswitchfalse 277 | * \@reversemarginfalse 278 | * (1in=72.27pt=25.4mm, 1cm=28.453pt) 279 | 280 | 281 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/context/base/mkii\supp-pd 282 | f.mkii" 283 | [Loading MPS to PDF converter (version 2006.09.02).] 284 | \scratchcounter=\count112 285 | \scratchdimen=\dimen133 286 | \scratchbox=\box57 287 | \nofMPsegments=\count113 288 | \nofMParguments=\count114 289 | \everyMPshowfont=\toks28 290 | \MPscratchCnt=\count115 291 | \MPscratchDim=\dimen134 292 | \MPnumerator=\count116 293 | \makeMPintoPDFobject=\count117 294 | \everyMPtoPDFconversion=\toks29 295 | ) 296 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/oberdiek\epstopdf-b 297 | ase.sty" 298 | Package: epstopdf-base 2016/05/15 v2.6 Base part for package epstopdf 299 | 300 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\infwarer 301 | r.sty" 302 | Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) 303 | ) 304 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/oberdiek\grfext.sty 305 | " 306 | Package: grfext 2016/05/16 v1.2 Manage graphics extensions (HO) 307 | 308 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\kvdefine 309 | keys.sty" 310 | Package: kvdefinekeys 2016/05/16 v1.4 Define keys (HO) 311 | 312 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\ltxcmds. 313 | sty" 314 | Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) 315 | ))) 316 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/oberdiek\kvoptions. 317 | sty" 318 | Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) 319 | 320 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\kvsetkey 321 | s.sty" 322 | Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) 323 | 324 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\etexcmds 325 | .sty" 326 | Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) 327 | 328 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\ifluatex 329 | .sty" 330 | Package: ifluatex 2016/05/16 v1.4 Provides the ifluatex switch (HO) 331 | Package ifluatex Info: LuaTeX not detected. 332 | )))) 333 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/generic/oberdiek\pdftexcm 334 | ds.sty" 335 | Package: pdftexcmds 2019/07/25 v0.30 Utility functions of pdfTeX for LuaTeX (HO 336 | ) 337 | Package pdftexcmds Info: LuaTeX not detected. 338 | Package pdftexcmds Info: \pdf@primitive is available. 339 | Package pdftexcmds Info: \pdf@ifprimitive is available. 340 | Package pdftexcmds Info: \pdfdraftmode found. 341 | ) 342 | Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 343 | 38. 344 | Package grfext Info: Graphics extension search list: 345 | (grfext) [.pdf,.png,.jpg,.mps,.jpeg,.jbig2,.jb2,.PDF,.PNG,.JPG,.JPE 346 | G,.JBIG2,.JB2,.eps] 347 | (grfext) \AppendGraphicsExtensions on input line 456. 348 | ) 349 | LaTeX Font Info: Try loading font information for U+msa on input line 18. 350 | 351 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsfonts\umsa.fd" 352 | File: umsa.fd 2013/01/14 v3.01 AMS symbols A 353 | ) 354 | LaTeX Font Info: Try loading font information for U+msb on input line 18. 355 | 356 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/amsfonts\umsb.fd" 357 | File: umsb.fd 2013/01/14 v3.01 AMS symbols B 358 | ) 359 | Underfull \hbox (badness 10000) in paragraph at lines 21--28 360 | 361 | [] 362 | 363 | 364 | Underfull \hbox (badness 10000) in paragraph at lines 21--28 365 | 366 | [] 367 | 368 | 369 | Underfull \hbox (badness 10000) in paragraph at lines 40--43 370 | 371 | [] 372 | 373 | 374 | Underfull \hbox (badness 10000) in paragraph at lines 45--49 375 | 376 | [] 377 | 378 | 379 | Underfull \hbox (badness 10000) in paragraph at lines 50--52 380 | 381 | [] 382 | 383 | LaTeX Font Info: Try loading font information for U+bbm on input line 52. 384 | ("C:\Users\User\AppData\Local\Programs\MiKTeX 2.9\tex/latex/bbm\ubbm.fd" 385 | File: ubbm.fd 1999/03/15 V 1.2 Font definition for bbm font - TH 386 | ) 387 | Underfull \hbox (badness 10000) in paragraph at lines 52--54 388 | 389 | [] 390 | 391 | [1 392 | 393 | {C:/Users/User/AppData/Local/MiKTeX/2.9/pdftex/config/pdftex.map}] 394 | Underfull \hbox (badness 10000) in paragraph at lines 60--63 395 | 396 | [] 397 | 398 | 399 | Underfull \hbox (badness 10000) in paragraph at lines 70--71 400 | 401 | [] 402 | 403 | 404 | Underfull \hbox (badness 10000) in paragraph at lines 73--74 405 | 406 | [] 407 | 408 | 409 | Underfull \hbox (badness 10000) in paragraph at lines 76--77 410 | 411 | [] 412 | 413 | 414 | Underfull \hbox (badness 10000) in paragraph at lines 83--84 415 | 416 | [] 417 | 418 | 419 | Underfull \hbox (badness 10000) in paragraph at lines 93--95 420 | 421 | [] 422 | 423 | [2] 424 | Underfull \hbox (badness 10000) in paragraph at lines 105--107 425 | 426 | [] 427 | 428 | 429 | Underfull \hbox (badness 10000) in paragraph at lines 135--137 430 | 431 | [] 432 | 433 | 434 | Underfull \hbox (badness 10000) in paragraph at lines 138--139 435 | 436 | [] 437 | 438 | [3] (ProlemSet7_solustion.aux) ) 439 | Here is how much of TeX's memory you used: 440 | 3100 strings out of 492895 441 | 43908 string characters out of 3124202 442 | 124353 words of memory out of 3000000 443 | 6959 multiletter control sequences out of 15000+200000 444 | 10986 words of font info for 45 fonts, out of 3000000 for 9000 445 | 1141 hyphenation exceptions out of 8191 446 | 41i,12n,32p,350b,199s stack positions out of 5000i,500n,10000p,200000b,50000s 447 | 464 | Output written on ProlemSet7_solustion.pdf (3 pages, 138074 bytes). 465 | PDF statistics: 466 | 69 PDF objects out of 1000 (max. 8388607) 467 | 0 named destinations out of 1000 (max. 500000) 468 | 1 words of extra memory for PDF output out of 10000 (max. 10000000) 469 | 470 | -------------------------------------------------------------------------------- /PS7/ProlemSet7_solustion.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS7/ProlemSet7_solustion.pdf -------------------------------------------------------------------------------- /PS7/ProlemSet7_solustion.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/hoangnguyen7699/StatisticsForApplication_solution/965f4893ea7785c3abda81f19cca3dc7b78f8123/PS7/ProlemSet7_solustion.synctex.gz -------------------------------------------------------------------------------- /PS7/ProlemSet7_solustion.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt]{article} 2 | 3 | \usepackage[margin=1in]{geometry} 4 | \usepackage{bbm} 5 | \usepackage{amsmath,amsthm,amssymb, graphicx, multicol, array} 6 | 7 | \newcommand{\N}{\mathbb{N}} 8 | \newcommand{\Z}{\mathbb{Z}} 9 | 10 | \newenvironment{problem}[2][Problem]{\begin{trivlist} 11 | \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} 12 | 13 | 14 | 15 | \begin{document} 16 | \title{Problem Set 2} 17 | \author{Hoang Nguyen, Huy Nguyen} 18 | \maketitle 19 | 20 | \begin{problem}{1} 21 | QQ-plots \\ 22 | \\ 23 | QQ-Plot 1: Laplace distribution with parameter $\sqrt{2}$ (heavier-heavier) \\ 24 | QQ-Plot 2: Uniform distribution on [-$\sqrt{3}$, $\sqrt{3}$] (lighter-lighter) \\ 25 | QQ-Plot 3: standard Gaussian distribution \\ 26 | QQ-Plot 4: exponential distribution with parameter 1 (lighter-heavier)\\ 27 | QQ-Plot 5: e Cauchy distribution (heavier-heavier)\\ 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | \end{problem} 36 | 37 | \begin{problem}{2} 38 | \item 1. Mark as update 39 | 40 | \item 2. \\ 41 | Let A(t) be the cdf for $U_i's$ \\ 42 | We have :\\ 43 | \[A(t)= \mathbb{P}(U_i \leq t)=\mathbb{P}\big(F(X_i)\leq t\big)\] 44 | Because F is increasing function: 45 | \[A(t)= \mathbb{P}\big(X_i\leq F^{-1}(t)\big)=F\big(F^{-1}(t) \big)=t\] 46 | Notice that $0\leq t \leq 1$ \\ 47 | Hence, distributions of the $U_i's$ is Uniform(0,1)\\ 48 | Do similarly we get distributions of the $V_i's$ is also Uniform(0,1) \\ 49 | 50 | \item 3.\\ 51 | a) \\ 52 | \[T_{n,m}=\sup_{t\in \mathbb{R}}\big|F_n(t)-G_m(t) \big|=\sup_{t\in \mathbb{R}}\big|\frac{1}{n} \sum_{i=1}^{n}\mathbbm{1}(X_i < t) -\frac{1}{m} \sum_{i=1}^{m}\mathbbm{1}(Y_i < t) \big|\] 53 | b) \\ 54 | \[T_{n,m}=\sup_{t\in \mathbb{R}}\big|F_n(t)-G_m(t) \big|=\sup_{t\in \mathbb{R}}\big|\frac{1}{n} \sum_{i=1}^{n}\mathbbm{1}(X_i < t) -\frac{1}{m} \sum_{i=1}^{m}\mathbbm{1}(Y_i < t) \big|\] 55 | 56 | F, G are increasing function 57 | \[\Rightarrow T_{n,m}= \sup_{t\in \mathbb{R}}\big|\frac{1}{n} \sum_{i=1}^{n}\mathbbm{1}\big(F(X_i) < F(t)\big) -\frac{1}{m} \sum_{i=1}^{m}\mathbbm{1}\big(G(Y_i) < G(t)\big) \big|\] 58 | \[=\sup_{t\in \mathbb{R}}\big|\frac{1}{n} \sum_{i=1}^{n}\mathbbm{1}\big(U_i < F(t)\big) -\frac{1}{m} \sum_{i=1}^{m}\mathbbm{1}\big(V_i < G(t)\big) \big|\] 59 | $H_0$ is true $\Rightarrow$ F(t)=G(t)=x ($0 \leq x \leq 1$): 60 | \[T_{n,m}=\sup_{0 \leq x \leq 1}\big|\frac{1}{n} \sum_{i=1}^{n}\mathbbm{1}\big(U_i < x\big) -\frac{1}{m} \sum_{i=1}^{m}\mathbbm{1}\big(V_i < x\big) \big|\] 61 | c)\\If H0 is true, the joint distribution of the n + m random variables $U_1,....,U_n,V_1,...,V_m$ is the joint distribution of the n + m Uniform(0,1)\\ 62 | d) \\ 63 | \[T_{n,m}=\sup_{0 \leq x \leq 1}\big|\frac{1}{n} \sum_{i=1}^{n}\mathbbm{1}\big(U_i < x\big) -\frac{1}{m} \sum_{i=1}^{m}\mathbbm{1}\big(V_i < x\big) \big|\] 64 | \[=\sup_{0 \leq x \leq 1}\big|\frac{1}{n} \sum_{i=1}^{n}Ber\big(\mathbb{P}(U_i < x)\big) -\frac{1}{m} \sum_{i=1}^{m}Ber\big(\mathbb{P}(V_i < x)\big) \big|\] 65 | \[=\sup_{0 \leq x \leq 1}\big|\frac{1}{n} \sum_{i=1}^{n}Ber\big(x\big) -\frac{1}{m} \sum_{i=1}^{m}Ber\big(x\big) \big|\] 66 | 67 | Hence, $T_{n,m}$ is pivotal\\ 68 | f) By CLT, we have 69 | \[\sqrt{n}\big(F_n(t)-F(t)\big) \xrightarrow[n \mapsto\infty ]{(d)} \mathbb{N}\big(0, F(t)(1-F(t)) \big) \] 70 | \[\Leftrightarrow F_n(t)-F(t)\xrightarrow[n \mapsto\infty ]{(d)} \mathbb{N}\big(0,\frac{F(t)(1-F(t))}{n} \big) \]\\ 71 | 72 | Similarly, we have: 73 | \[Q_m(t)-Q(t)\xrightarrow[m \mapsto\infty ]{(d)} \mathbb{N}\big(0,\frac{Q(t)(1-Q(t))}{m} \big) \]\\ 74 | 75 | From this, if $H_0$ is true ($F(t)=Q(t)$), by subtracting 2 equations above, we have: 76 | \[F_n(t)-Q_m(t) \xrightarrow[n,m \mapsto\infty ]{(d)} \mathbb{N}\big(0, \frac{F(t)(1-F(t))}{n} +\frac{Q(t)(1-Q(t))}{m} \big)\]\\ 77 | 78 | $H_0 $ is true: 79 | \[\Leftrightarrow F_n(t)-Q_m(t) \xrightarrow[n,m \mapsto\infty ]{(d)} \mathbb{N}\big(0,\sqrt{\frac{m+n}{mn}}F(t)(1-F(t)) \big)\] 80 | \[\Leftrightarrow \sqrt{\frac{mn}{m+n}}\big( F_n(t)-Q_m(t) \big) \xrightarrow[n,m \mapsto\infty ]{(d)} \mathbb{N}\big(0,F(t)(1-F(t)) \big)\] 81 | \[\Rightarrow\sqrt{\frac{mn}{m+n}} \sup_{t\in \mathbb{R}}\big| F_n(t)-Q_m(t) \big| \xrightarrow[n,m \mapsto\infty ]{(d)} \sup_{t\in \mathbb{R}} \big|\mathbb{B}(F(t))\big| \] 82 | 83 | where $\mathbb{B}$ is a Brownian bridge on [0,1]\\ 84 | 85 | Define a test $T_n=\sqrt{\frac{mn}{m+n}} \sup_{t\in \mathbb{R}}\big| F_n(t)-Q_m(t) \big| $ , by Donsker's theorem, if $H_0$ is true, then $T_n \xrightarrow[n,m \mapsto\infty ]{(d)} Z$ where X has a known distribution 86 | 87 | 88 | \end{problem} 89 | 90 | \begin{problem}{3} 91 | \item 1. 92 | Mark as update 93 | \item 2. 94 | We have: $R_i$, $R_j$ take the values 1,2,..n\\ 95 | 96 | With a,b=1,2,..n : 97 | \[\mathbb{P}(R_i=a, R_j=b)= (n-2)!\big(\frac{n-2}{n} \big)^{n-2} \] 98 | \[\mathbb{P}(R_i=a)= \frac{1}{n}\] 99 | \[\mathbb{P}(R_j=b)= \frac{1}{n}\] 100 | \[\Rightarrow \mathbb{P}(R_i=a, R_j=b)\neq \mathbb{P}(R_i=a)\mathbb{P}(R_j=b) \] 101 | 102 | Hence, $R_1,...R_{n}$ are not independent 103 | 104 | 105 | \item 3. 106 | We have: \\ 107 | 108 | $R_i$ takes the values in (1,2,..,n) 109 | \[\Rightarrow \mathbb{P}(R_i=a)=\begin{cases} \frac{1}{n} & \mbox{if } a\mbox{ =1,2,..,n} \\ 0, & \mbox{Otherwise } \end{cases}\] 110 | \[\Rightarrow R \backsim Uniform(1, n)\] 111 | 112 | Similarly: 113 | \[\Rightarrow Q \backsim Uniform(1, n)\] 114 | 115 | Hence, R, Q do not depend on distribution of $X_i$'s, $Y_i$'s 116 | 117 | \item 4. 118 | If $H_0$ is true: 119 | \[\mathbb{P}(R, Q)=\frac{1}{n^2}\] 120 | \[\mathbb{P}(R)=\frac{1}{n}\] 121 | \[\mathbb{P}(Q)=\frac{1}{n}\] 122 | \[\Rightarrow \mathbb{P}(R, Q)=\mathbb{P}(R)\mathbb{P}(Q)\] 123 | 124 | Hence, ($R_1$, $R_2$,...,$R_n$) and ($Q_1$, $Q_2$,...,$Q_n$) are independent 125 | 126 | \item 5. 127 | If $H_0$ is true, ($R_1$, $R_2$,...,$R_n$, $Q_1$, $Q_2$,...,$Q_n$) are the iid Uniform(1,n) distributon $\Rightarrow$ joint distribution of them does not depend on the distribution of the 128 | original sample 129 | 130 | \item 6. 131 | \[T_n=\frac{\sum_{i=1}^{n}(R_i-\bar{R_n})(Q_i-\bar{Q_n})}{\sqrt{\sum_{i=1}^{n}(R_i-\bar{R_n})^2\sum_{i=1}^{n}(Q_i-\bar{Q_n})^2}}\] 132 | \[T_n=\frac{\sum_{i=1}^n R_iQ_i -\sum_{i=1}^n \bar{R_n}(R_i+Q_i)+\sum_{i=1}^n \bar{R_i}\bar{Q_i}}{\sqrt{\sum_{i=1}^{n}(R_i-\bar{R_n})^2\sum_{i=1}^{n}(Q_i-\bar{Q_n})^2}}\] 133 | \[T_n=\frac{12}{n(n^2-1)}\sum_{i=1}^{n} R_iQ_i- \frac{3(n+1)}{n-1}\] 134 | 135 | \item 7. 136 | From question 3 we have R, Q do not depend on distribution of $X_i$'s, $Y_i$'s\\ 137 | 138 | $\Rightarrow$ $R_i '$ and $Q_i '$ are also Uniform(1, n)\\ 139 | 140 | $\Rightarrow S_n$ is the same distribution as $T_n$ 141 | 142 | 143 | 144 | 145 | \end{problem} 146 | 147 | \end{document} -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Solution Manual For Statistic For Applications (MIT) 2 | 3 | 4 | 5 | ### Instruction 6 | This is a Solution Manual for the famous course from MIT [Statistic For Application](https://ocw.mit.edu/courses/mathematics/18-650-statistics-for-applications-fall-2016/index.htm). 7 | 8 | The Manual is written in LaTex, and I do not omit other files generated by Tex, so that not only you can view the pdf file for solution but also others file generated by Tex. I will keep update the solutions as my learning process goes on. 9 | 10 | There are some problems that I am not sure, which have been marked by "Waiting for update" in the solution manual. If you want to provide a solution for these unsolving problems, have any question, or come up with better ideas about any problem in the manual, please feel free to contact with me 11 | 12 | I want to address several issues here: (1) Currently I still do not get the permission of Prof. Philippe Rigollet to upload this solution manual online. Therefore, maybe I will withdraw this solution maunal someday if it becomes illegal. (2) This solution manual is only meant to be used as personal study reference, **please do not take it for other purposes, e.g., upload to other websites or publications**. 13 | 14 | ### Read 15 | If you want to view only pdf files, you can clone this repository and find pdf file in each folders (There are 11 folders corresponding to 11 problem sets). 16 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet1.pdf) questions and [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/PS1/ProblemSet1.pdf) solution manual file for Problem set 1. 17 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet2.pdf) questions and [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/PS2/Problem_set2.pdf) solution manual file for Problem set 2. 18 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet3.pdf) questions and [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/PS3/ProblemSet3.pdf) solution manual file for Problem set 3. 19 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet4.pdf) questions and [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/PS4/ProbemSet4.pdf) solution manual file for Problem set 4 20 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet5.pdf) questions and [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/PS5/ProblemSet5.pdf) solution manual file for Problem set 5. 21 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet6.pdf) questions and [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/PS6/ProblemSet6.pdf) solution manual file for Problem set 6 22 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet7.pdf) questions and [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/PS7/ProlemSet7_solustion.pdf) solution manual file for Problem set 7 23 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet8.pdf) questions for Problem set 8 24 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet9.pdf) questions for Problem set 9 25 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet10.pdf) questions and [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/PS10/PropblemSet10.pdf) solution manual file for Problem set 10 26 | 1. [PDF](https://github.com/hoangnguyen7699/StatisticsForApplication_solution/blob/master/MIT18_650F16_PSet11.pdf) questions for Problem set 11 27 | 28 | ### Contribute 29 | This solution manual is work-in-progress so any kind of contributions would be appreciated. There are a few ways you can contribute to make it better: 30 | 1. Add/edit questions 31 | 1. Add/edit answers 32 | 1. Others 33 | 34 | 35 | 36 | ### Acknowledgments 37 | 38 | 1. Special thanks to [Huy Nguyen] (https://github.com/nguyehuy). He provides some analytical soultions for problems in this course's assigments. 39 | 40 | ** Star if you like :) ** 41 | --Hoang Nguyen-- 42 | --------------------------------------------------------------------------------