├── 123.pdf ├── Notes for Mathematics.pdf ├── Notes for Mathematics.synctex.gz ├── Notes for Mathematics.sin.gnuplot ├── Notes for Mathematics.tan-example.gnuplot ├── README.md ├── preface.tex ├── preface.aux ├── Notes for Mathematics.aux ├── algebra.aux ├── probility.aux ├── Notes for Mathematics.toc ├── analysis.aux ├── Notes for Mathematics.tex ├── Notes for Mathematics.out ├── probility.tex └── algebra.tex /123.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Hhhyouy/Notes-of-mathematics/HEAD/123.pdf -------------------------------------------------------------------------------- /Notes for Mathematics.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Hhhyouy/Notes-of-mathematics/HEAD/Notes for Mathematics.pdf -------------------------------------------------------------------------------- /Notes for Mathematics.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Hhhyouy/Notes-of-mathematics/HEAD/Notes for Mathematics.synctex.gz -------------------------------------------------------------------------------- /Notes for Mathematics.sin.gnuplot: -------------------------------------------------------------------------------- 1 | set table "Notes for Mathematics.sin.table"; set format "%.5f" 2 | set samples 100.0; plot [x=-5:5] sin(x) 3 | -------------------------------------------------------------------------------- /Notes for Mathematics.tan-example.gnuplot: -------------------------------------------------------------------------------- 1 | set table "Notes for Mathematics.tan-example.table"; set format "%.5f" 2 | set samples 100.0; plot [x=-3.141:3.141] [-3:3]tan(x) 3 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | 本来这篇前言或者说README洋洋洒洒落笔千言，最后删减到现在的三言两语，我还是决定把最纯粹的内容奉献给大家。我写这份笔记，一方面是个人学习的总结，便于不断地复习回顾；另一方面也是希望能够给那些在学习数学上遇到困难的同道中人一些微不足道的帮助；最后，比较详尽的公式让那些有教学、科研需要的朋友们可以到源文件内直接复制粘贴，大大减轻了他们的工作量，也能更加美观。本笔记错误、疏漏在所难免，希望大家谅解。 2 | 使用时直接对Notes for Mathematics.tex编译一次即可。为了方便预览，本人将事先编译过的文件上传到此，也可以直接打开Notes for Mathematics.pdf阅读。 3 | -------------------------------------------------------------------------------- /preface.tex: -------------------------------------------------------------------------------- 1 | \section{前言} 2 | 本来洋洋洒洒落笔千言，最后删减到现在的三言两语，我还是决定把最纯粹的内容奉献给大家。我写这份笔记，一方面是个人学习的总结，便于不断地复习回顾；另一方面也是希望能够给那些在学习数学上遇到困难的同道中人一些微不足道的帮助；最后，比较详尽的公式让那些有教学、科研需要的朋友们可以到源文件内直接复制粘贴，大大减轻了他们的工作量，也能更加美观。本笔记错误、疏漏在所难免，希望大家谅解。 3 | 4 | 这份笔记是由\LaTeX{}编写的，我认为\LaTeX{}有更自由的方式来实现我偏好的设计，同时其对于数学公式的排版也是诸如Word、Indesign等无法媲美的。在笔记的体例上，大致是正文和特殊环境的组合。特殊环境中，定义被环境左侧的视觉引导线包裹，证明结束的标志是实心的小正方形$\blacksquare $，定理和例题的分别是灯泡和图钉。这样，正文和特殊环境总是能够区分开来。同时，我定义了一系列的命令如：\jie \zheng \txe{注：} 等等来方便录入。笔记的内容参考了本科数学教材、考研教辅资料和真题等等。 5 | 6 | 另外，本笔记内容(包括.tex源文件)全部开源，并遵循CC协议。 7 | 8 | \hfill 可达可达 9 | 10 | \hfill 2023年1月2日 -------------------------------------------------------------------------------- /preface.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \@writefile{toc}{\contentsline {section}{\numberline {1}前言}{5}{section.1}\protected@file@percent } 4 | \@setckpt{preface}{ 5 | \setcounter{page}{6} 6 | \setcounter{equation}{0} 7 | \setcounter{enumi}{0} 8 | \setcounter{enumii}{0} 9 | \setcounter{enumiii}{0} 10 | \setcounter{enumiv}{0} 11 | \setcounter{footnote}{0} 12 | \setcounter{mpfootnote}{0} 13 | \setcounter{part}{0} 14 | \setcounter{section}{1} 15 | \setcounter{subsection}{0} 16 | \setcounter{subsubsection}{0} 17 | \setcounter{paragraph}{0} 18 | \setcounter{subparagraph}{0} 19 | \setcounter{figure}{0} 20 | \setcounter{table}{0} 21 | \setcounter{parentequation}{0} 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overlay unbroken={ 134 | \draw[Blue,line width=0.5mm] (title.west) -- ([yshift=\tcboxedtitleheight/2]frame.north west) -- (frame.south west) -- (frame.south east);}, 135 | overlay first={ 136 | \draw[Blue,line width=0.5mm] (title.west) -- ([yshift=\tcboxedtitleheight/2]frame.north west) -- (frame.south west);}, 137 | overlay middle={ 138 | \draw[Blue,line width=0.5mm] (frame.north west)--(frame.south west);}, 139 | overlay last={ 140 | \draw[Blue,line width=0.5mm] (frame.north west)--(frame.south west)--(frame.south east);} 141 | } 142 | 143 | \newtcolorbox[auto counter,number within=section]{definition}[1][]{enhanced,frame hidden,colback=white,coltitle=Blue,fonttitle=\bfseries,title={定义~\thetcbcounter},detach title,before upper={\tcbtitle\quad},after title={\quad#1},breakable,sharp corners,borderline west={1mm}{0mm}{Cream_3}} 144 | 145 | \newtcolorbox{case}{enhanced,colframe=Lake_Blue!10,colback=Lake_Blue!10,sharp corners,breakable,borderline west={1mm}{0mm}{Lake_Blue}} 146 | 147 | \newtcolorbox[auto counter,number within=section]{liti}[1][]{colback=white,coltitle=Blue,frame hidden,breakable,before skip=1cm,title=错题回顾 \thetcbcounter,enhanced,attach boxed title to top left,sharp corners,fonttitle=\LARGE\heiti, 148 | boxed title style={empty,colframe=Blue,size=minimal,overlay={\draw[Blue,line width=2pt]([yshift=2pt]frame.north west)--([yshift=2pt]frame.north east);}}, 149 | overlay unbroken={ 150 | \draw[Blue,line width=1pt] ([yshift=2pt]title.north west) -- ([yshift=\tcboxedtitleheight+2pt]frame.north east); 151 | \draw[Blue,line width=1pt] (frame.south west)--(frame.south east);}, 152 | overlay first={ 153 | \draw[Blue,line width=1pt] ([yshift=2pt]title.north west) -- ([yshift=\tcboxedtitleheight+2pt]frame.north east);}, 154 | overlay last={ 155 | \draw[Blue,line width=1pt] (frame.south west)--(frame.south east);} 156 | } 157 | 158 | \newtcbox{\txe}{on line,arc=1pt,colback=Blue!10,colframe=white,colupper=Blue,boxrule=0pt,boxsep=0pt,left=5pt,right=3pt,top=2pt,bottom=2pt} 159 | 160 | \newtcolorbox[auto counter]{examp}[1][]{enhanced,frame hidden,colback=white,coltitle=Blue,fonttitle=\bfseries,title={例~\thetcbcounter},detach title,before upper={\tcbtitle},after title={\quad#1},breakable,sharp corners,size=minimal, 161 | overlay unbroken={ 162 | \node [yshift=0.5em] at (frame.south east) {\Large\color{Blue}\tiPinOutline};}, 163 | overlay last={ 164 | \node [yshift=0.5em] at (frame.south east) {\Large\color{Blue}\tiPinOutline};}} 165 | 166 | \newtcolorbox[auto counter]{ttheorem}[1][]{enhanced,frame hidden,colback=white,coltitle=Blue,title={\bfseries 定理~\thetcbcounter},detach title,before upper={\tcbtitle},after title={\heiti#1\quad},breakable,sharp corners,beforeafter skip=0.5em,size=minimal, 167 | overlay unbroken={ 168 | \node [yshift=0.5em] at (frame.south east) {\Large\color{Blue}\tiLightbulb};}, 169 | overlay last={ 170 | \node [yshift=0.5em] at (frame.south east) {\Large\color{Blue}\tiLightbulb};}} 171 | 172 | %%%%%%%%%%%%%%%%%%Color Definition%%%%%%%%%%%%%%%% 173 | \definecolor{Light_Yellow}{HTML}{CF7F19} 174 | \definecolor{Light_Green}{HTML}{007C63} 175 | \definecolor{Lake_Blue}{HTML}{7192C5} 176 | \definecolor{Blue}{HTML}{2e9ce9} 177 | %Nature 178 | \definecolor{Nature_1}{HTML}{F4FCD9} 179 | \definecolor{Nature_2}{HTML}{C5D8A4} 180 | \definecolor{Nature_3}{HTML}{BB9981} 181 | \definecolor{Nature_4}{HTML}{534340} 182 | %Vintage 183 | \definecolor{Vintage_1}{HTML}{E5E3C9} 184 | \definecolor{Vintage_2}{HTML}{B4CFB0} 185 | \definecolor{Vintage_3}{HTML}{94B49F} 186 | \definecolor{Vintage_4}{HTML}{789395} 187 | %Cream 188 | \definecolor{Cream_1}{HTML}{F7FBFC} 189 | \definecolor{Cream_2}{HTML}{D6E6F2} 190 | \definecolor{Cream_3}{HTML}{B9D7EA} 191 | \definecolor{Cream_4}{HTML}{769FCD} 192 | 193 | \title{Notes for Mathematics$^{\, \ccbync}$} 194 | \author{可达可达} 195 | \date{} 196 | 197 | \begin{document} 198 | \includepdf{123.pdf} 199 | \newpage 200 | \maketitle 201 | \tableofcontents 202 | 203 | \zihao{-4} 204 | \include{preface} 205 | \include{analysis} 206 | \include{algebra} 207 | \include{probility} 208 | 209 | \end{document} 210 | 211 | 212 | -------------------------------------------------------------------------------- /Notes for Mathematics.out: -------------------------------------------------------------------------------- 1 | \BOOKMARK [1][-]{section.1}{\376\377\122\115\212\000}{}% 1 2 | \BOOKMARK [1][-]{section.2}{\376\377\232\330\173\111\145\160\133\146}{}% 2 3 | \BOOKMARK [2][-]{subsection.2.1}{\376\377\122\035\173\111\145\160\133\146\143\320\143\010}{section.2}% 3 4 | \BOOKMARK [3][-]{subsubsection.2.1.1}{\376\377\136\070\165\050\116\011\211\322\121\375\145\160\121\154\137\017}{subsection.2.1}% 4 5 | \BOOKMARK [3][-]{subsubsection.2.1.2}{\376\377\143\222\122\027\176\304\124\010}{subsection.2.1}% 5 6 | \BOOKMARK 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[3][-]{subsubsection.4.5.5}{\376\377\123\072\225\364\117\060\213\241}{subsection.4.5}% 83 84 | \BOOKMARK [3][-]{subsubsection.4.5.6}{\376\377\120\107\213\276\150\300\232\214}{subsection.4.5}% 84 85 | \BOOKMARK [1][-]{section.5}{\376\377\226\104\137\125}{}% 85 86 | -------------------------------------------------------------------------------- /probility.tex: -------------------------------------------------------------------------------- 1 | 2 | \section{概率论与数理统计} 3 | \subsection{随机事件及其概率} 4 | \subsubsection{事件间的关系} 5 | \begin{enumerate} 6 | \item 包含 7 | 8 | 如果事件$A$发生必然导致事件$B$发生,则称事件$B$包含事件$A$,记作$A\subset B$ 9 | 10 | 符号$\subset $可以看成小于号$<$,表示范围的大小,方便记忆. 11 | \item 和 12 | 13 | 两个事件$A$与$B$中至少有一个发生,称为事件$A$与$B$的和,记作$A+B$或$A\cup B$ 14 | \item 积 15 | 16 | 两个事件$A$与$B$同时发生,称为事件$A$与$B$的积,记作$AB$或$A\cap B$ 17 | \item 互不相容 18 | 19 | 如果事件$A$与$B$不能同时发生,则称事件$A$与$B$互不相容或互斥,此时满足$AB=\varnothing $,若A与B互不相容,则有\begin{gather*} 20 | P(AB)=0,\\ 21 | P(A+B)=P(A)+P(B),\\ 22 | A\subset \overline{B} 23 | \end{gather*} 24 | \item 对立（逆） 25 | 26 | 如果两个事件$A$与$B$满足$A+B=\varOmega,AB=\varnothing $,则称事件$A$是事件$B$的对立事件或逆事件,$\overline{A}=B$或$\overline{B}=A$.要注意的是: 27 | \begin{gather*} 28 | P(\overline{A})=1-P(A)\\ 29 | \text{对立}\Rightarrow \text{互不相容}\\ 30 | \overline{A}\cdot \overline{B}\neq \overline{AB},\overline{ABC}\neq \overline{A}+\overline{B}+\overline{C}\\ 31 | \end{gather*} 32 | \end{enumerate} 33 | 34 | 事件之间只有一部分重合并不代表不独立,判断事件间的独立性只能通过具体计算得出.当然包含事件必然不是独立的. 35 | \subsubsection{事件的运算律} 36 | \begin{enumerate} 37 | \item 分配律 38 | \[(A+B)C=AC+BC\] 39 | \item 对偶原理 40 | \[\overline{A+B}=\overline{A}\cdot\overline{B}\] 41 | \[\overline{AB}=\overline{A}+\overline{B}\] 42 | \item 若$A \implies B$,则$\overline{B} \implies \overline{A}$. 43 | \item 若$A+B\subset C$,则$AB\subset C$. 44 | \item $AB\subset A+B$ 45 | \end{enumerate} 46 | \subsubsection{随机事件的概率公式} 47 | 设$A$,$B$为任意两个随机事件 48 | \begin{itemize} 49 | \item 若$A\supset B$,则有 50 | \begin{gather*} 51 | P(A-B)=P(A)-P(B)\\ 52 | P(AB)=P(B) 53 | \end{gather*} 54 | \item 若$A,B$互斥则有 55 | \begin{equation*} 56 | P(A+B)=P(A)+P(B) 57 | \end{equation*} 58 | \item 减法公式 59 | \[P(A \overline{B})=P(A-B)=P(A-AB)=P(A)-P(AB)\] 60 | \item 加法公式 61 | \begin{gather*} 62 | P(A+B)=P(A)+P(B)-P(AB)\\ 63 | P(A+B+C)=P(A)+P(B)+P(C)\\ 64 | \phantom{P(A+B+C)=}-P(AB)-P(AC)-P(BC)+P(ABC) 65 | \end{gather*} 66 | \end{itemize} 67 | 68 | \begin{ttheorem} 69 | 若事件$A$与$B$独立,则$A$与$\overline{B} $,$\overline{A} $与$B$,$\overline{A} $与$\overline{B} $也相互独立. 70 | \end{ttheorem} 71 | 72 | 设事件$A,B$相互独立,则有： 73 | \begin{enumerate} 74 | \item $P(AB)=P(A)\times P(B)$ 75 | \item $P(A\vert B)=P(A)$ 76 | \item $P(A\vert B)=P(A\vert \overline{B})$ 77 | \item $P(A+B)=1-P(\overline{A+B} )=1-P(\overline{A}\cdot\overline{B} )=1-P(\overline{A} )P(\overline{B} )$ 78 | \item $P(A-B)=P(A\overline{B} )=P(A)P(\overline{B} )$ 79 | \end{enumerate} 80 | 81 | \begin{definition}[条件概率] 82 | \[P(A\vert B)=\frac{P(AB)}{P(B)}\] 83 | 84 | 称$P(A\vert B)$为事件$B$发生的条件下$A$发生的条件概率 85 | \end{definition} 86 | 87 | 条件概率的性质: 88 | \begin{itemize} 89 | \item $P(\overline{A}\vert B)=1-P(A\vert B)$ 90 | \item $P(A_1+A_2\vert B)=P(A_1\vert B)+P(A_2\vert B)-P(A_1A_2\vert B)$ 91 | \item 乘法定理：设$P(B)>0$,则有 92 | \[P(AB)=P(B)\cdot P(A\vert B)\] 93 | 设$P(A)>0$,则有 94 | \[P(AB)=P(A)\cdot P(B\vert A)\] 95 | \begin{gather*} 96 | P(A_1A_2A_3\cdots A_n)=P(A_1)P(A_2\vert A_1)P(A_3\vert A_1A_2)\\ 97 | \phantom{P(A_1A_2A_3\cdots A_n)=}\cdots P(A_n\vert A_1A_2A_3\cdots A_{n-1}) 98 | \end{gather*} 99 | \end{itemize} 100 | \begin{ttheorem}[(全概率公式)] 101 | 设事件$A_1,A_2,\dots,A_n$是样本空间$\varOmega $的一个分割,他们两两不相容且$\sum_{i=1}^{\infty} A_i=\varOmega$,则对任意一个事件$B$有 102 | \begin{equation*} 103 | P(B)=\sum_{i=1}^{\infty} P(BA_i)=\sum_{i=1}^{\infty} P(A_i)P(B\vert A_i). 104 | \end{equation*} 105 | \end{ttheorem} 106 | \begin{theorem}[Bayes公式] 107 | 设$A_1,A_2,\dotsc,A_i,\dotsc$是样本空间$\varOmega $的一个完备事件组,且$P(A_i)>0,i=1,2,\dotsc$.对任一事件$B$,若$P(B)>0$,则有 108 | \begin{equation*} 109 | P(A_i\vert B)=\frac{P(A_i)P(B\vert A_i)}{\sum\limits^{\infty}_{i=1}P(A_i)P(B\vert A_i)} 110 | \end{equation*} 111 | 112 | 可以写成 113 | \begin{equation*} 114 | P(A_i B)=\frac{P(A_iB)}{\sum\limits^{\infty}_{i=1}P(A_iB)} 115 | \end{equation*} 116 | \end{theorem} 117 | \begin{liti} 118 | \begin{enumerate}[label=\protect\enumlabel{\thetcbcounter\arabic*}, leftmargin=0mm] 119 | \item （2006·湖北·14）安排5名歌手的演出顺序时,要求某名歌手不第一个出场,另一名歌手不最后一个出场,不同排法的种数是（用数字作答）. 120 | 121 | 答：分类讨论. 122 | 123 | 当不最后一个出场的歌手第一个出场时,原先不能第一个出场的歌手肯定不会出现在第一个.此时排法有$A_4^4=24$种. 124 | 125 | 当不最后一个出场的歌手不第一个出场时,第一个位置只能有三种选择,第二个位置有三种选择,第三个位置有两种选择,第四个位置有一种选择,最后一个位置有三种选择,此时排法有$C_3^1\times A_3^3 \times C_3^1(3\times 3\times 2\times 1\times 3)=54$种. 126 | 127 | 所以总排法有$24+54=78$种. 128 | 129 | \item （2010·湖北·8）现安排甲、乙、丙、丁、戊5名同学参加上海世博会志愿者服务活动,每人从事翻译、导游、礼仪、司机四项工作之一,每项工作至少有一人参加.甲、乙不会开车但能从事其他三项工作,丙、丁、戊都能胜任四项工作,则不同安排方案的种数是（\quad）. 130 | 131 | \onech{54}{90}{126}{152} 132 | 答：不平均分配问题$+$分类讨论 133 | 134 | 五名同学安排四项工作,一定会有一项工作多出一名同学,也就是三项一名同学的工作和一项两名同学的工作的排法问题.特殊的地方在于,甲乙是不会开车的,所以如果多出甲乙是不能给他们安排司机的.所以要进行分类讨论. 135 | 136 | 多出一名同学的工作是司机,此时,两名司机肯定只能在丙丁戊中选,甲乙从事另外三项工作,一共有$A_3^3\times C_3^2=18$种. 137 | 138 | 多出一名同学的工作不是司机,此时,司机只需要在丙丁戊中选一名,一共有$A_3^3\times C_4^2\times C_3^1=108$种. 139 | 140 | 所以一共有$18+108=126$种. 141 | 142 | \item （2009·湖北·5）将甲、乙、丙、丁四名学生分到三个不同的班,每个班至少分到一名学生,且甲、乙两名学生不能分到同一个班,则不同分法的种数为（\quad）. 143 | 144 | \onech{18}{24}{30}{36} 145 | 答：不平均分配问题 146 | 147 | 甲、乙、丙、丁四名学生分到三个不同的班,一定会有一个班多出一个学生,排列就变成了两个一个人的班级和一个两个人的班级的排法问题.每个班至少分到一名学生的分法一共有$A_3^3\times C_4^2=36$种,因为甲乙两名学生不能分到同一个班,所以四个人中选择两人同班的分法要减少一种,此时一共有$A_3^3\times (C_4^2-1)=30$种. 148 | \end{enumerate} 149 | \end{liti} 150 | 151 | \subsection{常见的几种分布} 152 | 本章常用的几个积分公式： 153 | \begin{gather*} 154 | \int_{-\infty}^{+\infty} \mathrm{e} ^{-x^2} \,\mathrm{d}x =\sqrt{\pi}\\ 155 | \int_{0}^{+\infty} x^ne^{-x} \,\mathrm{d}x =n!\\ 156 | \sum_{k=0}^{\infty} k\cdot \frac{\lambda^k}{k!}e^{-\lambda}=\lambda\\ 157 | \sum_{k=0}^{\infty} \frac{\lambda^k}{k!}e^{-\lambda}=1 158 | \end{gather*} 159 | 第三个式子用来计算Poisson分布的期望. 160 | 161 | 分布函数的性质： 162 | \begin{enumerate} 163 | \item $0 \leqslant F(x) \leqslant 1$ 164 | \item $\lim_{x \to -\infty} F(x)=0,\lim_{x \to +\infty} F(x)=1$ 165 | \item $F(x)$是单调非减函数 166 | \item $F(x)$是右连续的,即$P(X \leqslant x)=F(x)=P(X5$和$n(1-p)>5$时用正态分布近似比较好. 193 | 194 | 如果随机变量序列$X_1,X_2,\dots,X_n$独立同分布,$X_i\sim B(1,p),i=1,2,\dots,n$,那么$\sum_{i=1}^{\infty} X_i$服从二项分布,$\sum_{i=1}^{\infty} X_i \sim B(n,p)$. 195 | 196 | 若$X\sim B(n_1,p),Y\sim B(n_2,p)$且$X,Y$相互独立,则$X+Y\sim B(n_1+n_2,p)$ 197 | \item 超几何分布 198 | 199 | $P(X=k)=\dfrac{C^k_MC_{N-M}^{n-k}}{C_N^n}$,若满足$N\to \infty,M \to \infty,\dfrac{M}{N}\to p$则有$P(X=k)=\dfrac{C^k_MC_{N-M}^{n-k}}{C_N^n}=C^k_np^k(1-p)^{n-k}$ 200 | 201 | \item 泊松分布 202 | 203 | 随机变量X可取一切非负整数值,$P(X=k)=\dfrac{\lambda^k}{k!}e^{-\lambda},k=0,1,2,3,\dots,\lambda>0$,泊松分布$EX=\lambda$,$DX=\lambda$. 204 | 205 | 注意$k$取非负整数,$\lambda>0$. 206 | 207 | 若$X\sim P(\lambda_1),Y\sim P(\lambda_2)$且$X,Y$相互独立,则$X+Y\sim P(\lambda_1+\lambda_2)$ 208 | 209 | \item 几何分布 210 | 211 | $X\sim G(p)$满足$P(X=k)=(1-p)^{k-1}\cdot p,k=1,2,3,\dots$ 212 | \end{enumerate} 213 | 214 | \subsubsection{一维连续型随机变量} 215 | 概率密度函数的性质： 216 | \begin{enumerate} 217 | \item $0 \leqslant f(x) <1$ 218 | \item $\int_{-\infty}^{+\infty} f(x) \,\mathrm{d}x =1$ 219 | \item 对任意实数$x_1 < x_2$,有$P(x_10$,是常数.$EX=\dfrac{1}{\lambda}$,$DX=\dfrac{1}{\lambda^2}$. 247 | 248 | \item 正态分布$X\sim N(\mu,\sigma^2)$ 249 | 250 | 对应的概率密度函数和分布函数分别为 251 | \[f(x)=\frac{1}{\sqrt{2\pi }\sigma}\mathrm{e} ^{-\frac{(x-\mu )^2}{2\sigma ^2}},-\infty0,\sigma>0\] 252 | 和$\displaystyle F(x)=\frac{1}{\sqrt{2\pi }\sigma} \int_{-\infty}^{x} \mathrm{e} ^{-\frac{(t-\mu )^2}{2\sigma ^2}} \,dt ,-\infty0$,$\varPhi (x)$的值查表获得；当$x<0$时,$\varPhi (x)=1-\varPhi (-x)$. 261 | 262 | 正态分布的$EX=\mu,DX=\sigma^2$,标准正态分布的$EX=0,DX=1$. 263 | 264 | 若$X \sim N(\mu,\sigma^2)$,则$X+a\sim N(\mu+a,\sigma^2),aX\sim N(a\mu,a^2\sigma^2)$. 265 | 266 | 正态分布的对称轴为$x=\mu$,最大值为$\frac{1}{\sqrt{2\pi}\sigma}$. 267 | \end{enumerate} 268 | \subsubsection{二维随机变量} 269 | 二维离散型随机变量的分布函数为 270 | \begin{equation*} 271 | \sum_{x_i\leqslant x}\sum_{y_j\leqslant y}P(X=x_i,Y=y_i)=\sum_{x_i\leqslant x}\sum_{y_j\leqslant y}p_{ij} 272 | \end{equation*} 273 | 274 | 二维连续型随机变量的概率密度函数为$P(X=x,Y=y)=f(x,y)$ 275 | 276 | 二维连续型随机变量的分布函数为 277 | \begin{equation*} 278 | P(X \leqslant x,Y \leqslant y)=F(x,y)=\int_{-\infty}^{x}\int_{-\infty}^{y}f(u,v) \,\mathrm{d} u\,\mathrm{d} v 279 | \end{equation*} 280 | 他的边缘分布函数为$P(X \leqslant x)=F(x,+\infty)=\int_{-\infty}^{x} \int_{-\infty}^{+\infty}f(u,y) \,\mathrm{d} y \,\mathrm{d}u$.和$P(Y \leqslant y)=F(+\infty,y)=\int_{-\infty}^{y} \int_{-\infty}^{+\infty}f(x,v) \,\mathrm{d} x \,\mathrm{d}v $,边缘概率密度为$f_X(x)=F_X'(x)=\int_{-\infty}^{+\infty} f(x,y) \,\mathrm{d}y ,f_Y(y)=F_Y'(y)=\int_{-\infty}^{+\infty} f(x,y) \,\mathrm{d}x $ 281 | 282 | 二维随机变量$(X,Y)$落在某一平面的概率,就是对他在该平面的概率密度函数求积分.这与求分布函数不同,分布函数是从负无穷开始积分. 283 | 284 | 二维分布函数的性质： 285 | \begin{enumerate} 286 | \item $0 \leqslant F(x,y) \leqslant 1$ 287 | \item $F(-\infty,y)=F(x,-\infty)=F(-\infty,-\infty)=0$ 288 | \item $F(+\infty,+\infty)=1$ 289 | \item $F(x,y)$单调不减 290 | \item $F(x,y)$右连续 291 | \item $P(a0$,则称 344 | \begin{equation*} 345 | f_{X\vert Y}(x\vert y)=\frac{f(x,y)}{f_Y(y)} 346 | \end{equation*} 347 | 为在$Y=y$条件下$X$的条件概率密度. 348 | \end{definition} 349 | 350 | 随机变量的独立性和条件概率密度要结合事件的运算加深理解.若$X,Y$相互独立,则$P(X=x_i,Y=y_i)=P(Y=y_i)P(X=x_i\vert Y=y_i)=P(Y=y_i)P(X=x_i)$. 351 | 352 | \subsubsection{二维随机变量函数的分布} 353 | 求二维随机变量函数的分布有两种方法,对于简单的函数$Z=X+Y$可以采用卷积公式,这个方法要求随机变量$X,Y$相互独立.对于更复杂的函数而言采用的方法,与求二维随机变量落在某一平面的概率相类似,先求他的分布函数$F_Z(z)=P(Z \leqslant z)$,求导后就得到了二维随机变量函数的概率密度函数$f_Z(z)=[F_Z(z)]'$. 354 | 355 | 由于$f_Z(z) \geqslant 0$,使用分布函数法要讨论$Z(X,Y)$所组成的平面区域和概率密度函数$f(x,y)$非零的区域之间的情况.而使用连续卷积公式同样要讨论$f_X(x)f_Y(z-x)>0$的情况. 356 | \begin{ttheorem}[(连续卷积公式)] 357 | 设$X,Y$是两个相互独立的连续型随机变量,他们的概率密度分别为$f_X(x)$和$f_Y(y)$,则$Z=X+Y$的概率密度为 358 | \begin{equation*} 359 | f_Z(z)=\int_{-\infty}^{+\infty} f_X(x)f_Y(z-x) \,\mathrm{d}x , 360 | \end{equation*} 361 | 或 362 | \begin{equation*} 363 | f_Z(z)=\int_{-\infty}^{+\infty} f_X(z-y)f_Y(y) \,\mathrm{d}y , 364 | \end{equation*} 365 | \end{ttheorem} 366 | 367 | \begin{ttheorem}[(离散卷积公式)] 368 | 设$X,Y$是两个相互独立的连续型随机变量,他们的概率密度分别为$P(X=x_i)$和$P(y=y_j)$,则$Z=X+Y$的概率密度为 369 | \begin{equation*} 370 | P(Z=z_k)=\sum_i P(X=x_i)P(Y=z_k-x_i) 371 | \end{equation*} 372 | \end{ttheorem} 373 | 374 | \subsubsection{极值分布} 375 | 376 | 设随机变量$X_1,X_2,\dotsc,X_n$相互独立,分布函数分别为$F_i(x),i=1,2,\dotsc,n$. 377 | 378 | 最大值$X^*_n=\max\{X_1,X_2,\dotsc,X_n\} $的分布函数为 379 | \begin{equation*} 380 | P(x^*_n \leqslant x)= \prod _{i=1}^{n}F_i(x) 381 | \end{equation*} 382 | 如果是独立同分布,则有$F_{X_n^*}(x)=[F(x)]^n$ 383 | 384 | 最小值$X_1^*=\min\{X_1,X_2,\dotsc,X_n\}$的分布函数为 385 | \begin{equation*} 386 | P(x_1^* \leqslant x)=1-\prod_{i=1}^{n}\left[1-F_i(x)\right] 387 | \end{equation*} 388 | 如果是独立同分布,则有$F_{X_1^*}(x)=1-[1-F(x)]^n$. 389 | \subsection{随机变量的数字特征} 390 | \subsubsection{离散型随机变量的数字特征} 391 | \begin{definition} 392 | 设离散型随机变量$X$的分布律为$P(X=x_i)=p_i,i=1,2,\dots,n.$若$\sum^\infty_{i=1}\left\lvert x_i \right\rvert p_i<+\infty,$则称 393 | 394 | \[ 395 | EX=\sum^\infty_{i=1}x_ip_i 396 | \] 397 | 为随机变量$X$的算术数学期望,简称期望或均值. 398 | \end{definition} 399 | 400 | 若离散型随机变量$X$的分布律为$P(X=x_i)=p_i(i=1,2,3,\dots)$,$g(X)$是$X$的某一函数,则$g(X)$的数学期望\[ 401 | E[g(X)]=\sum^{}_{i}g(x_i)p_i\] 402 | 403 | 若离散型随机向量$(X,Y)$的联合分布律为$P(X=x_i,Y=y_i)=p_ij,i,j=1,2,3\dots$,$Z=g(X,Y)$是$X,Y$二元函数,则有\[ 404 | EZ=E[g(X,Y)]=\sum^{}_{i}\sum^{}_{j}g(x_i,y_i)p_{ij}\] 405 | 如果要算$EX$,$EY$不必求出边缘密度函数,可用下式\[ 406 | EX=\sum^{}_{i}\sum^{}_{j}x_ip_{ij}\] 407 | 408 | \begin{definition} 409 | 设$X$为随机变量,若$E(X-EX)^2$存在,则称$E(X-EX)^2$为$X$的方差,记为$DX$或$VarX$,即\[DX=E\left[(X-EX)^2\right] \]称方差$DX$的算数根$\sqrt{DX}$为$X$的标准差或均方差,记作$\sigma_X$或简记为$\sigma$. 410 | \end{definition} 411 | 412 | 若$X$是离散型随机变量,分布律为$P(X=x_i)=p_i(i=1,2,3\dots)$,则\[DX=\sum^{}_{i}(x_i-EX)^2p_i\] 413 | \subsubsection{连续型随机变量的数字特征} 414 | \begin{definition} 415 | 设连续型随机变量$X$的密度函数为$f(x)$,若$\int_{-\infty}^{+\infty}\left\lvert x \right\rvert f(x) \,\mathrm{d} x<+\infty$ 则称 416 | \[ 417 | EX=\int_{-\infty}^{+\infty}xf(x) \,\mathrm{d} x 418 | \]为随机变量X的算术数学期望,简称期望或均值. 419 | \end{definition} 420 | 421 | 若连续型随机变量$X$的概率密度函数为$f(x)$,$g(X)$是$X$的某一函数,则$g(X)$的数学期望\[ 422 | E[g(X)]=\int_{-\infty}^{+\infty}g(x)f(x) \,\mathrm{d} x \] 423 | 若连续型随机变量的联合概率密度函数为$f(x,y)$,则\[ 424 | EZ=E[g(X,Y)]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}g(x,y)f(x,y) \,\mathrm{d} x\,\mathrm{d} y\]如果要算$EX$,$EY$不必求出边缘密度函数,可用下式\[ 425 | EX=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}xf(x,y) \,\mathrm{d} x\,\mathrm{d} y\] 426 | 427 | 若$X$是连续型随机变量,密度函数为$f(x)$,则\[DX=\int_{-\infty}^{-\infty}(x-EX)^2f(x) \,\mathrm{d} x \] 428 | 429 | \subsubsection{随机变量数字特征的联系} 430 | 431 | 方差的计算常用以下公式 432 | \begin{equation*} 433 | DX=EX^2-(EX)^2 434 | \end{equation*} 435 | 这个公式同时给出了计算$EX^2$的方法.如果已知了随机变量的期望与方差,要求$EX^2$,只要计算$DX+(EX)^2$. 436 | 437 | \begin{examp}{$X\sim N(\mu,\sigma^2)$,求$EX^2$.} 438 | 439 | \jie $EX^2=DX+(EX)^2=\sigma^2+\mu^2$ 440 | \end{examp} 441 | 442 | \begin{definition} 443 | 设$(X,Y)$为二维随机向量,若$E[(X-EX)(Y-EY)]$存在,则称此期望为$X,Y$的协方差,记为$Cov(X,Y)$,即\[ 444 | Cov(X,Y)=E[(X-EX)(Y-EY)]=E(XY)-EXEY. 445 | \] 446 | \end{definition} 447 | \begin{definition} 448 | 设$(X,Y)$为二维随机向量,且$X,Y$的方差$\sigma_X\sigma_Y$均为正,则称$\dfrac{Cov(X,Y)}{\sigma_X\sigma_Y}$为$X$与$Y$的相关系数,记为$\rho _{XY}$或简记为$\rho $,即\[ 449 | \rho _{XY}=\frac{Cov(X,Y)}{\sigma_X\sigma_Y}=\frac{E[(X-\mu_X)(Y-\mu_Y)]}{\sigma_X\sigma_Y}. 450 | \] 451 | \end{definition} 452 | 期望的性质 453 | \begin{enumerate} 454 | \item 设$c$为常数,$Ec=c$. 455 | \item $E(cX)=cE(X)$. 456 | \item $E(X+Y)=EX+EY$. 457 | \item 设$X,Y$相互独立,则有$E(XY)=EXEY$. 458 | \end{enumerate} 459 | 460 | 方差的性质 461 | \begin{enumerate} 462 | \item 设$c$为常数,$Dc=0$. 463 | \item $DX \geqslant 0$ 464 | \item $D(cX)=c^2D(X)$. 465 | \item $D(X+c)=DX$ 466 | \item 设$X,Y$是两个随机变量,则\[D(X\pm Y)=DX+DY\pm 2Cov(X,Y).\]特别地,若$X,Y$相互独立,则有$D(X\pm Y)=DX+DY$. 467 | \end{enumerate} 468 | 469 | 协方差计算公式 470 | \begin{enumerate} 471 | \item $Cov(X,X)=DX.$ 472 | \item $Cov(X,Y)=Cov(Y,X).$ 473 | \item $Cov(aX,bY)=abCov(X,Y)$,$ab$为任意常数. 474 | \item $Cov(a,X)=0$,$ab$为任意常数. 475 | \item $Cov(X_1+X_2,Y)=Cov(X_1,Y)+Cov(X_2,Y).$ 476 | \end{enumerate} 477 | 478 | 设随机变量$X,Y$的相关系数为$\rho_{XY}$,则$\left\lvert \rho_{XY}\right\rvert \leqslant 1$ 479 | \begin{ttheorem}[（独立性与不相关性）] 480 | 若方差均大于零的随机变量$X,Y$相互独立,则必不相关.但$X,Y$不相关不一定相互独立. 481 | \end{ttheorem} 482 | \begin{align*} 483 | XY\text{不相关}\iff &Cov(X,Y)=0\\ 484 | \iff &\rho_{XY}=0\\ 485 | \iff &E(XY)=EXEY\\ 486 | \iff &D(X\pm Y)=DX+DY\\ 487 | \end{align*} 488 | \begin{ttheorem}[（Markov不等式）] 489 | 设$X$是一个非负的随机变量且期望存在,则对任意$a>0$,有\[ 490 | P(X\geqslant a)\leqslant \frac{EX}{a}.\] 491 | \end{ttheorem} 492 | \begin{ttheorem}[（Chebyshev不等式）] 493 | 设随机变量$X$的期望和方差都存在,则对任意常数$\varepsilon >0$,有\[ 494 | P(\left\lvert X-EX \right\rvert \geqslant \varepsilon )\leqslant \frac{DX}{\varepsilon ^2}.\]或\[ 495 | P(\left\lvert X-EX \right\rvert < \varepsilon )\geqslant 1- \frac{DX}{\varepsilon ^2}.\] 496 | \end{ttheorem} 497 | \begin{ttheorem}[（Cauchy-Schwarz不等式）] 498 | 设$XY$是任意两个随机变量,若$X,Y$的方差都存在,则有\[ 499 | [E(X,Y)]^2\leqslant EX^2EY^2\] 500 | \end{ttheorem} 501 | \subsection{大数定律和中心极限定理} 502 | \begin{theorem}[Bernoulli大数定律] 503 | 设$\mu_n$为$n$重Bernoulli实验中事件$A$发生的次数,$p$为每次实验中$A$发生的概率,则对任意的$\varepsilon >0$,有\[ 504 | \lim_{n \to \infty}P(\left\lvert \frac{\mu_n}{n}-p \right\rvert< \varepsilon )=1 \] 505 | \end{theorem} 506 | 该定理说明了当$n$足够大时,频率等于概率. 507 | 508 | \begin{definition} 509 | 设$Y_1,Y_2,\dots,Y_n,\dots$为随机变量序列,$a$为常数,如果对任意的$\varepsilon >0$,有\[ 510 | \lim_{n \to \infty}P(\left\lvert Y_n-a \right\rvert< \varepsilon )=1 \]则称${{Y_n}}$依概率收敛于$a$,记作$Y_n\xrightarrow[]{P} a$ 511 | \end{definition} 512 | \begin{theorem}[Chebyshev大数定律] 513 | 设$X_1,X_2,\dots,X_n,\dots$是两两不相关的随机变量序列,且方差都是一致有界的,即存在某一常数$C$,使得$DX_i0$,都有 514 | \begin{equation*} 515 | \label{chebyshev} 516 | \lim_{n \to \infty}P\left(\left\lvert \frac{1}{n}\sum^{n}_{i=1}X_i-\frac{1}{n}\sum^{n}_{i=1}EX_i \right\rvert< \varepsilon \right)=1 517 | \end{equation*} 518 | \end{theorem} 519 | \begin{theorem}[Markov大数定律] 520 | 若随机变量序列$X_1,X_2,\dots,X_n,\dots$满足条件\[\lim_{n \to \infty}\frac{1}{n^2}D\left(\sum^{n}_{i=1}X_i\right)=0\],则该序列服从大数定律,即对任意$\varepsilon >0$,\eqref{chebyshev}成立. 521 | \end{theorem} 522 | \begin{theorem}[Khinchin大数定律] 523 | 设$X_1,X_2,\dots,X_n,\dots$是相互独立且同分布的随机变量序列,$EX_i=\mu,i=1,2,\dots,$则序列$X_1,X_2,\dots,X_n,\dots$服从大数定律,即对任意$\varepsilon >0$,有 524 | \begin{equation*} 525 | \lim_{n \to \infty}P\left(\left\lvert \frac{1}{n}\sum^{n}_{i=1}X_i-\mu \right\rvert< \varepsilon \right)=1 526 | \end{equation*} 527 | \end{theorem} 528 | 该定理说明了当$n$足够大时,均值等于期望. 529 | 530 | \begin{theorem}[Lindeberg-Levy中心极限定理] 531 | 设$X_1,X_2,\dots,X_n,\dots$是相互独立且同分布的随机变量序列,$EX_i=\mu,DX_i=\sigma^2(0<\sigma^2<+\infty),i=1,2,\dots,$则对任意的$x\in \mathbb{R}$ ,有 532 | \begin{equation*} 533 | \lim_{n \to \infty}P\left(\frac{\sum\limits^{n}_{i=1}X_i-n\mu}{\sqrt{n}\sigma} \leqslant x \right)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}} \,\mathrm{d} t =\varPhi (x) 534 | \end{equation*} 535 | \end{theorem} 536 | \begin{theorem}[De Moivre-Laplace中心极限定理] 537 | 设$\mu_n$为$n$重Bernoulli实验中事件$A$发生的次数,$p$为每次实验中$A$发生的概率,则对任意的$x\in \mathbb{R}$,有 538 | \begin{equation*} 539 | \lim_{n \to \infty}P\left(\frac{\mu_n-np}{\sqrt{np(1-p)}}\leqslant x \right)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}} \,\mathrm{d} t =\varPhi (x) 540 | \end{equation*} 541 | \end{theorem} 542 | \subsection{统计估计方法} 543 | \subsubsection{总体} 544 | \begin{enumerate} 545 | \item 设$X$是随机变量,把$E(X^k),k=1,2,\dots$称为$X$的$k$阶原点矩. 546 | \item 设$X$是随机变量,把$E\left[(X-EX)^k\right] ,k=1,2,\dots$称为$X$的$k$阶中心矩. 547 | \end{enumerate} 548 | \subsubsection{样本} 549 | 设$X_1,X_2,\dotsc,X_n$为取自某总体$X$的样本,定义样本均值为 550 | \begin{equation*} 551 | \bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i. 552 | \end{equation*} 553 | 样本方差为 554 | \begin{equation*} 555 | S^2=\frac{1}{n-1}\sum^{n}_{i=1}(X_i-\bar{X})^2 556 | \end{equation*} 557 | 样本标准差为 558 | \begin{equation*} 559 | S=\sqrt{S^2}=\sqrt{\frac{1}{n-1}\sum^{n}_{1}(X_i-\bar{X})^2} 560 | \end{equation*} 561 | 样本的k阶原点矩为 562 | \begin{equation*} 563 | A_k=\frac{1}{n}\sum^{n}_{i=1}X_i^k,k=1,2,\dots 564 | \end{equation*} 565 | 样本的k阶中心矩为 566 | \begin{equation*} 567 | B_k=\frac{1}{n}\sum^{n}_{i=1}(X_i-\bar{X})^k,k=1,2,\dots 568 | \end{equation*} 569 | 当$k=2$时,习惯上将$B_2$记为$S_n^2$.满足$B_2=A_2-\bar{X}^2$,即$\frac{1}{n}\sum^{n}_{i=1}(X_i-\bar{X})^2=\frac{1}{n}\sum^{n}_{i=1}X_i^2-\bar{X}^2$. 570 | \begin{prf}[$\frac{1}{n}\sum^{n}_{i=1}(X_i-\bar{X})^2=\frac{1}{n}\sum^{n}_{i=1}X_i^2-\bar{X}^2$] 571 | \begin{align*} 572 | \frac{1}{n}\sum^{n}_{i=1}(X_i-\bar{X})^2&=\frac{1}{n}\sum^{n}_{i=1}(X_i^2-2X_i\bar{X}+\bar{X}^2) \\ 573 | &=\frac{1}{n}\sum^{n}_{i=1}X_i^2-\frac{2\bar{X}}{n}\sum^{n}_{i=1}X_i+\frac{1}{n}\sum^{n}_{i=1}\bar{X}^2=\frac{1}{n}\sum^{n}_{i=1}X_i^2-2\bar{X}^2+\bar{X}^2 \\ 574 | &=\frac{1}{n}\sum^{n}_{i=1}X_i^2-\bar{X}^2. 575 | \end{align*} 576 | \end{prf} 577 | \begin{prf}[$\sum^{n}_{i=1}(x_i-\bar{x})^2=\sum^{n}_{i=1}x_i(x_i-\bar{x})$] 578 | \begin{align*} 579 | \sum^{n}_{i=1}(x_i-\bar{x})^2&=\sum^{n}_{i=1}(x^2_i-2x_i\bar{x}+\bar{x}^2) \\ 580 | &=\sum^{n}_{i=1}x^2_i-2\bar{x}\sum^{n}_{i=1}x_i+n\bar{x}^2=\sum^{n}_{i=1}x^2_i-n\bar{x}^2 =\sum^{n}_{i=1}x_i^2-\bar{x}\sum^{n}_{i=1}x_i\\ 581 | &=\sum^{n}_{i=1}x_i(x_i-\bar{x}). 582 | \end{align*} 583 | \end{prf} 584 | \begin{prf}[$\sum^{n}_{i=1}(x_i-\bar{x})(y_i-\bar{y})=\sum^{n}_{i=1}x_i(y_i-\bar{y})$] 585 | \begin{align*} 586 | \sum^{n}_{i=1}(x_i-\bar{x})(y_i-\bar{y})&=\sum^{n}_{i=1}(x_iy_i-\bar{y}x_i-\bar{x}y_i+\bar{x}\bar{y}) \\ 587 | &=\sum^{n}_{i=1}x_iy_i-\bar{y}\sum^{n}_{i=1}x_i-\bar{x}\sum^{n}_{i=1}y_i+n\bar{x}\bar{y}=\sum^{n}_{i=1}x_iy_i-n\bar{x}\bar{y} \\ 588 | &=\sum^{n}_{i=1}x_i(y_i-\bar{y}). 589 | \end{align*} 590 | \end{prf} 591 | \begin{prf}[$\sum^{n}_{i=1}(X_i-\bar{X})^2=\sum^{n}_{i=1}(X_i-\mu)^2-n(\bar{X}-\mu)^2$] 592 | 593 | \end{prf} 594 | \subsubsection{抽样分布} 595 | 抽样的样本属于相互独立的随机变量,如果总体的期望和方差分别为$EX=\mu,DX=\sigma^2$,那么样本均值$\bar{X}$和样本方差$S^2$是相互独立的随机变量且满足以下特征： 596 | \begin{enumerate} 597 | \item $E\bar{X}=\mu$ 598 | \item $D\bar{X}=\frac{\sigma^2}{n}$ 599 | \item $ES^2=DX=\sigma^2$ 600 | \end{enumerate} 601 | 602 | \begin{definition} 603 | 设$X_1,X_2,\dotsc,X_n$独立同分布于标准正态分布$N(0,1)$,则$\chi^2=X_1^2+\dotsc+X_n^2$服从自由度为$n$的$\chi^2$分布,记为$\chi^2\sim\chi^2(n)$. 604 | \end{definition} 605 | 卡方分布的性质： 606 | \begin{enumerate} 607 | \item 设$\chi^2_1\sim \chi^2(n_1),\chi^2_2\sim \chi^2(n_2)$,并且$\chi^2_1,\chi^2_2$相互独立,则$\chi^2_1+\chi^2_2\sim \chi^2(n_1+n_2)$ 608 | \item 设$\chi^2\sim \chi^2(n)$则有$E(\chi^2)=n,D(\chi^2)=2n$ 609 | \end{enumerate} 610 | \begin{definition} 611 | 设$X\sim N(0,1),Y\sim \chi^2(n)$,且$XY$相互独立,则称随机变量 612 | \begin{equation*} 613 | t=\frac{X}{\sqrt{Y/n}} 614 | \end{equation*} 615 | 服从自由度为$n$的$t$分布,记为$t\sim t(n)$. 616 | \end{definition} 617 | 618 | $t_{1-\alpha}(n)=-t_\alpha(n)$. 619 | 620 | $t$分布的概率密度函数是偶函数. 621 | 622 | \begin{definition} 623 | 设$X\sim \chi^2(n_1),Y\sim \chi^2(n_2)$,且$XY$相互独立,则称随机变量 624 | \begin{equation*} 625 | F=\frac{X/n_1}{Y/n_2} 626 | \end{equation*} 627 | 服从自由度为$(n_1,n_2)$的$F$分布,记为$F\sim F(n_1,n_2)$. 628 | \end{definition} 629 | $\frac{1}{F}\sim F(n_2,n_1)$ 630 | 631 | 设$X_1,X_2,\dotsc,X_n$是来自正态总体$N(\mu,\sigma^2)$的样本,$\bar{X},S^2$分别是样本均值和样本方差,则有 632 | \begin{enumerate} 633 | \item \begin{equation*} 634 | \bar {X}\sim N(\mu,\frac{\sigma^2}{n}) 635 | \end{equation*} 636 | \item \begin{equation*} 637 | \frac{\bar{X}-\mu}{\sigma/\sqrt{n}} \sim N(0,1) 638 | \end{equation*} 639 | \item \begin{equation*} 640 | \frac{1}{\sigma^2}\sum^{n}_{i=1}(X_i-\mu)^2\sim \chi^2(n) 641 | \end{equation*} 642 | 643 | 对于该式,可以这样理解,由于$\dfrac{X_i-\mu}{\sigma}\sim N(0,1)$,故$\displaystyle \sum_{i=1}^{n} \left(\frac{X_i-\mu}{\sigma}\right) ^2=\frac{1}{\sigma^2}\sum^{n}_{i=1}(X_i-\mu)^2$是卡方分布的标准形式. 644 | \item \begin{equation} 645 | \dfrac{(n-1)S^2}{\sigma^2}\sim \chi^2(n-1); 646 | \end{equation} 647 | 648 | 对于该式,可以这样理解,$\displaystyle S^2=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2,\frac{(n-1)S^2}{\sigma^2}=\sum_{i=1}^{n}\frac{(X_i-\bar{X})^2}{\sigma^2}$ 649 | \item \begin{equation*} 650 | \frac{\bar{X}-\mu}{S/\sqrt{n}}\sim t(n-1) 651 | \end{equation*} 652 | 对于该式不是硬背下来的,而是构造出来的,考虑t分布的构造方法,易得 653 | \begin{equation} 654 | \frac{\dfrac{\bar{X}-\mu}{\sigma/\sqrt{n}} }{\sqrt{\dfrac{(n-1)S^2}{\sigma^2}/(n-1)}}=\frac{\sqrt{n}}{S}(\bar{X}-\mu)\sim t(n-1) 655 | \end{equation} 656 | \end{enumerate} 657 | 658 | \subsubsection{参数估计} 659 | 矩估计的原理是当样本数量无穷大时,由Khinchin大数定律,样本矩等于总体矩$EX^k=\frac{1}{n}\sum_{i=1}^nX_i^k$.所以总体有$k$个参数,就可以列出$k$个方程,计算总体和样本的$1\sim k$阶矩. 660 | 661 | 由于矩估计既可以采用原点矩,又可以采用中心矩,故具体操作时使用更简单的原点矩即可. 662 | 663 | \subsubsection{区间估计} 664 | 设已给定置信水平为$1-\alpha$,$X_1,X_2,\dotsc,X_n$是来自正态总体$N(\mu,\sigma^2)$的样本,$\bar{X},S^2$分别是样本均值和样本方差,则当$\sigma^2$已知时,$\mu$的置信区间为 665 | \begin{equation*} 666 | \left(\bar{X}\pm \frac{\sigma}{\sqrt{n}}z_{\frac{\alpha}{2}}\right) 667 | \end{equation*} 668 | 当$\sigma^2$未知时,$\mu$的置信区间为 669 | \begin{equation*} 670 | \left(\bar{X}\pm \frac{S}{\sqrt{n}}t_{\frac{\alpha}{2}}(n-1)\right) 671 | \end{equation*} 672 | 当$\mu$未知时,$\sigma^2$的置信区间为 673 | \begin{equation*} 674 | \left(\frac{(n-1)S^2}{\chi^2_{\frac{\alpha}{2}}(n-1)},\frac{(n-1)S^2}{\chi^2_{1-\frac{\alpha}{2}}(n-1)}\right) 675 | \end{equation*} 676 | \subsubsection{假设检验} 677 | 假设检验的拒绝域： 678 | \newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}} 679 | \begin{table}[] 680 | \begin{tabular}{ccccc} 681 | 原假设 $H_0$& 备择假设 $H_1$& 其他参数 & 检验统计量 & 拒绝域 \\ 682 | $\mu=\mu_0$ & $\mu\neq \mu_0$ & \multirow{3}{*}{$\sigma^2$已知} & \multirow{3}{*}{$Z=\frac{\bar{X}-\mu_0}{\sigma/\sqrt{n}}$} & $\left\lvert Z\right\rvert \geqslant z_{\frac{\alpha}{2}}$ \\ 683 | $\mu\leqslant \mu_0$ & $\mu>\mu_0$ & & & $Z\geqslant z_\alpha$ \\ 684 | $\mu\geqslant \mu_0$ & $\mu<\mu_0$ & & & $Z\leqslant -z_\alpha$ \\ 685 | $\mu=\mu_0$ & $\mu\neq \mu_0$ & \multirow{3}{*}{$\sigma^2$未知} & \multirow{3}{*}{$t=\frac{\bar{X}-\mu_0}{S/\sqrt{n}}$} & $\left\lvert t\right\rvert \geqslant t_{\frac{\alpha}{2}}(n-1)$ \\ 686 | $\mu\leqslant \mu_0$ & $\mu>\mu_0$ & & & $t\geqslant t_\alpha(n-1)$ \\ 687 | $\mu\geqslant \mu_0$ & $\mu<\mu_0$ & & & $t\leqslant -t_\alpha(n-1)$ \\ 688 | $\sigma^2=\sigma_0^2$ & $\sigma^2\neq \sigma_0^2$ & \multirow{3}{*}{$\mu$未知} & \multirow{3}{*}{$\chi^2=\frac{(n-1)S^2}{\sigma^2}$} & \tabincell{c}{$\chi^2\leqslant \chi^2_{1-\frac{\alpha}{2}}(n-1)$或\\$\chi^2\geqslant \chi^2_{\frac{\alpha}{2}}(n-1)$ }\\ 689 | $\sigma^2\leqslant \sigma_0^2$ & $\sigma^2> \sigma_0^2$ & & & $\chi^2\geqslant \chi^2_{\alpha}(n-1)$ \\ 690 | $\sigma^2\geqslant \sigma_0^2$ & $\sigma^2< \sigma_0^2$ & & & $\chi^2\leqslant \chi^2_{1-\alpha}(n-1)$ 691 | \end{tabular} 692 | \end{table} 693 | 694 | \clearpage 695 | \section{附录} 696 | 每天起床头件事,先背一遍展开式： 697 | \begin{align*} 698 | \mathrm{e}^x&=\sum^{\infty}_{n=0}\phantom{\frac{1}{n!}x^n}=\phantom{1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots},-\inftyp_s$且$p_k$在$p_s$前面,则称$p_k$与$p_s$形成一个逆序,一个排列中所有逆序的个数成为该排列的逆序数,记为$\tau (p_1p_2\dots p_n)$. 5 | \end{definition} 6 | \begin{definition}[$n$阶行列式] 7 | $n^2$个数排列成 8 | \begin{equation*} 9 | D=\left\lvert a_{ij}\right\rvert= 10 | \begin{vmatrix} 11 | a_{11} & \dots & a_{1n} \\ 12 | \vdots & \ddots & \vdots \\ 13 | a_{n1} & \dots & a_{nn} \\ 14 | \end{vmatrix} 15 | \end{equation*} 16 | 称为$n$阶行列式,其值为$\displaystyle D=\sum_{p_1\dots p_n} (-1)^{\tau (p_1p_2\dots p_n)}a_{1p_1}a_{2p_2}\dots a_{np_n}$.也称此式为$D$的展开式.其中$p_1p_2\dots p_n$为$1,2,\dots,n$的某个排列,$\displaystyle \sum_{p_1\dots p_n}$表示对所有排列求和. 17 | 18 | 其值也等于$\displaystyle D=\sum_{p_1\dots p_n} (-1)^{\tau (p_1p_2\dots p_n)}a_{p_11}a_{p_22}\dots a_{p_nn}$. 19 | \end{definition} 20 | 副对角型行列式 21 | $D= 22 | \begin{vmatrix} 23 | 0 & \dots & 0 & a_1 \\ 24 | 0 & \dots & a_2 & 0 \\ 25 | \vdots & \iddots & \vdots & \vdots \\ 26 | a_n & \dots & 0 & 0 \\ 27 | \end{vmatrix}$ 28 | 的结果为$(-1)^{\frac{n(n-1)}{2}}a_1a_2\dots a_n$. 29 | 30 | 范德蒙德行列式 31 | \begin{equation*} 32 | \begin{vmatrix} 33 | 1 & 1 & 1 & \dots & 1 \\ 34 | x_1 & x_2 & x_3 & \dots & x_n \\ 35 | x_1^2 & x_2^2 & x_3^2 & \dots & x_n^2 \\ 36 | \vdots & \vdots & \vdots & \ddots & \vdots \\ 37 | x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \dots & x_n^{n-1} \\ 38 | \end{vmatrix} 39 | =\prod _{1 \leqslant js$,则向量组$\alpha_1,\alpha_2,\dots,\alpha_r$必线性相关. 1124 | \end{ttheorem} 1125 | \begin{ttheorem} 1126 | 如果向量组$\alpha_1,\alpha_2,\dots,\alpha_r$可由向量组$\beta_1,\beta_2,\dots,\beta_s$线性表示,且向量组$\alpha_1,\alpha_2,\dots,\alpha_r$线性无关,则$r \leqslant s$. 1127 | \end{ttheorem} 1128 | 1129 | 求向量组秩的方法：将向量组中的向量看成矩阵的列向量组成一个矩阵,然后用初等变化法化为行阶梯型矩阵,就得出了向量组的秩,同时根据每行第一个不为零的数所在的列,还得出如何将向量组的剩余向量用极大无关组表示. 1130 | 1131 | 向量组等价,秩相等,但秩相等不能推出两个向量组等价.向量组$(\alpha_1,\alpha_2,\dots,\alpha_r)$若能线性表示向量组$(\beta_1,\beta_2,\dots,\beta_r)$,本质上就是线性方程组有解,$(\alpha_1,\alpha_2,\dots,\alpha_r)x=\beta_1,(\alpha_1,\alpha_2,\dots,\alpha_r)x=\beta_2,\dots$,所以等价条件是$r(\mathbf{A},\mathbf{B})=r(\mathbf{A})$.(增广矩阵的秩等于系数矩阵的秩.) 1132 | 1133 | \subsubsection{Schimidt正交化} 1134 | 1135 | \begin{definition} 1136 | 设有非零向量组$\alpha_1,\alpha_2,\dots,\alpha_s$,若他们两两正交,即$(\alpha_i,\alpha_j)=0,i,j=1,2,\dots,s$,则称该向量组为正交向量组. 1137 | \end{definition} 1138 | \begin{ttheorem} 1139 | 正交向量组必为线性无关向量组. 1140 | \end{ttheorem} 1141 | 设$\alpha_1,\alpha_2,\dots,\alpha_s$为线性无关向量组,则向量组$\beta_1,\beta_2,\dots,\beta_s$为正交向量组. 1142 | \begin{gather*} 1143 | \beta_1=\alpha_1,\\ 1144 | \beta_i=\alpha_i-\sum_{k=1}^{i-1} \frac{(\alpha_i,\beta_k)}{(\beta_k,\beta_k)}\beta_k,i=2,3,\dots,s 1145 | \end{gather*} 1146 | 亦即$\beta_1=\alpha_1,\beta_2=\alpha_2-\frac{(\alpha_2,\beta_1)}{(\beta_1,\beta_1)}\beta_1,\beta_3=\alpha_3-\frac{(\alpha_3,\beta_1)}{(\beta_1,\beta_1)}\beta_1-\frac{(\alpha_3,\beta_2)}{(\beta_2,\beta_2)}\beta_2,\dots$ 1147 | 1148 | 接下来再将向量组$\beta_1,\beta_2,\dots,\beta_s$单位化, 1149 | \begin{gather*} 1150 | \eta _i=\frac{\beta_i}{\left\lVert \beta_i \right\rVert }. 1151 | \end{gather*} 1152 | 1153 | \subsection{线性方程组} 1154 | \subsubsection{线性方程组解的存在性} 1155 | 1156 | \begin{ttheorem} 1157 | 一个一般的线性方程组 1158 | \begin{equation*} 1159 | \begin{cases} 1160 | a_{11}x_1+a_{12}x_2+\dotsm+a_{1n}x_n= b_1 \\ 1161 | a_{21}x_1+a_{22}x_2+\dotsm+a_{2n}x_n= b_2 \\ 1162 | \dots \\ 1163 | a_{m1}x_1+a_{m2}x_2+\dotsm+a_{mn}x_n= b_m \\ 1164 | \end{cases} 1165 | \end{equation*} 1166 | 有解的充分必要条件是系数矩阵$\mathbf{A}$的秩等于增广矩阵$\widetilde{\mathbf{A}} $的秩,即$r(\mathbf{A})=r(\widetilde{\mathbf{A}} )$.当$r=n$时,方程组有唯一解. 1167 | \end{ttheorem} 1168 | 1169 | 需要注意的是,若系数矩阵不是方阵,而是$m\times n$矩阵,那么$n$才代表变量个数,$m$代表方程的个数. 1170 | 1171 | \begin{theorem}[克拉默(Cramer)法则] 1172 | 对于一个一般的线性方程组 1173 | \begin{gather*} 1174 | \begin{cases} 1175 | a_{11}x_1+a_{12}x_2+\dotsm+a_{1n}x_n= b_1 \\ 1176 | a_{21}x_1+a_{22}x_2+\dotsm+a_{2n}x_n= b_2 \\ 1177 | \dots \\ 1178 | a_{m1}x_1+a_{m2}x_2+\dotsm+a_{mn}x_n= b_m \\ 1179 | \end{cases} 1180 | \end{gather*} 1181 | 当$m=n$时,如果系数行列式$\left\lvert \mathbf{A}\right\rvert \neq 0$,则方程组有唯一解： 1182 | \begin{equation*} 1183 | x_j=\frac{\left\lvert \mathbf{A}_j\right\rvert}{\left\lvert \mathbf{A}\right\rvert},j=1,2,\dots,n. 1184 | \end{equation*} 1185 | 1186 | $\left\lvert \mathbf{A}_j\right\rvert$是用常数列代替系数行列式中的第$j$列元素得到的行列式. 1187 | 1188 | 当常数列都为零时,该线性方程组称为齐次线性方程组.当$\left\lvert \mathbf{A}\right\rvert \neq 0$时,齐次线性方程组仅有零解. 1189 | 1190 | 齐次线性方程组有非零解的充分必要条件是其系数行列式等于零,即$\left\lvert \mathbf{A}\right\rvert =0$ 1191 | \begin{gather*} 1192 | \vert \mathbf{A} \vert \neq 0 \iff \text{$\mathbf{A}$可逆} \iff \mathbf{A}^{-1}=\frac{1}{\left\lvert \mathbf{A} \right\rvert}\mathbf{A}^*\\ 1193 | \iff \text{线性方程组有唯一解} \iff \text{对应的齐次线性方程组仅有零解} 1194 | \end{gather*} 1195 | \end{theorem} 1196 | 1197 | \begin{ttheorem} 1198 | 齐次线性方程组 1199 | \begin{equation*} 1200 | \begin{cases} 1201 | a_{11}x_1+a_{12}x_2+\dotsm+a_{1n}x_n= 0 \\ 1202 | a_{21}x_1+a_{22}x_2+\dotsm+a_{2n}x_n= 0 \\ 1203 | \dots \\ 1204 | a_{m1}x_1+a_{m2}x_2+\dotsm+a_{mn}x_n= 0 \\ 1205 | \end{cases} 1206 | \end{equation*} 1207 | 有非零解的充分必要条件是$r(\mathbf{A})0 1552 | \end{gather*} 1553 | 则称二次型为正定二次型,对应矩阵称为正定矩阵. 1554 | \end{definition} 1555 | 1556 | \begin{ttheorem} 1557 | 可逆线性变换不改变二次型的正定性. 1558 | \end{ttheorem} 1559 | 1560 | 这个定理给我们判断一个二次型的正定性提供了一个思路,只要将一个二次型化为标准型,若其平方项的系数都大于零,那么这个二次型就是正定二次型. 1561 | 1562 | \begin{ttheorem}[（$f$正定的充要条件）] 1563 | \begin{gather*} 1564 | f(x_1,x_2,\dots,x_n)=x^\mathrm{T}\mathbf{A}x\text{正定}\\ 1565 | \iff \mathbf{A} \text{的正惯性指数}p=n\\ 1566 | \iff \mathbf{A}\simeq \mathbf{I}, \text{即存在可逆阵$\mathbf{C}$,使得}\mathbf{C}^\mathrm{T}\mathbf{A}\mathbf{C}=\mathbf{I}.\text{(正定矩阵一定合同于单位矩阵)}\\ 1567 | \iff \mathbf{A}=\mathbf{D}^\mathrm{T}\mathbf{D},\text{其中$\mathbf{D}$是可逆阵}\\ 1568 | \iff \mathbf{A}\text{的全部特征值}\lambda_i>0,i=1,2,\dots,n\\ 1569 | \iff \mathbf{A}\text{的全部顺序主子式大于零},\\ 1570 | \text{即对}\mathbf{A}= 1571 | \begin{pmatrix} 1572 | a_{11} & a_{12} & \dots & a_{1n} \\ 1573 | a_{21} & a_{22} & \dots & a_{2n} \\ 1574 | \vdots & \vdots & \ddots & \vdots \\ 1575 | a_{n1} & a_{n2} & \dots & a_{nn} \\ 1576 | \end{pmatrix} 1577 | ,a_{11}>0,\begin{vmatrix} 1578 | a_{11} & a_{12} \\ 1579 | a_{21} & a_{22} \\ 1580 | \end{vmatrix} 1581 | >0,\dots, 1582 | \left\lvert \mathbf{A}\right\rvert >0 1583 | \end{gather*} 1584 | \end{ttheorem} 1585 | 1586 | 若二次型$f(x_1,x_2,\dots,x_n)=x^\mathrm{T}\mathbf{A}x$正定,则 1587 | \begin{enumerate} 1588 | \item $\mathbf{A}$的主对角元素$a_{ii}>0$ 1589 | \item $\mathbf{A}$的行列式$\left\lvert \mathbf{A}\right\rvert >0$ 1590 | \end{enumerate} 1591 | 1592 | 1593 | 1594 | --------------------------------------------------------------------------------