├── Obsidian └── .obsidian │ ├── app.json │ ├── appearance.json │ ├── themes │ └── AnuPpuccin │ │ └── manifest.json │ ├── core-plugins.json │ ├── core-plugins-migration.json │ └── workspace.json ├── Notes ├── Group Theory │ └── Group Theory.pdf ├── Applied Graph Theory │ ├── Images │ │ ├── MST.pdf │ │ ├── Prim.pdf │ │ ├── Cycle1.pdf │ │ ├── Path1.pdf │ │ ├── Walk1.pdf │ │ ├── Walks.pdf │ │ ├── BullGraph.pdf │ │ ├── Dijkstra.pdf │ │ ├── GNM(6,11).pdf │ │ ├── GNM(6,8).pdf │ │ ├── GNM(7,8).pdf │ │ ├── L(GNM(7,8)).pdf │ │ ├── Subgraphs-G.pdf │ │ ├── 6BlockGraph2.pdf │ │ ├── ShortestPaths.pdf │ │ ├── Subgraphs-H1.pdf │ │ ├── Subgraphs-H2.pdf │ │ ├── Subgraphs-H3.pdf │ │ ├── 6BlockGraph2-Blocks.pdf │ │ ├── L(G)Neighbourhood.pdf │ │ ├── BullGraph.ipe │ │ └── Subgraphs-H1.ipe │ ├── Applied Graph Theory.pdf │ ├── Applied Graph Theory (Small Screen).pdf │ ├── Applied Graph Theory.tex │ ├── Applied Graph Theory (Small Screen).tex │ ├── Preamble.tex │ └── Applied Graph Theory.out.ps ├── MAT 6102 - Graph Theory │ ├── Images │ │ ├── MST.pdf │ │ ├── Cycle1.pdf │ │ ├── Path1.pdf │ │ ├── Prim.pdf │ │ ├── Walk1.pdf │ │ ├── Walks.pdf │ │ ├── Dijkstra.pdf │ │ ├── GNM(6,8).pdf │ │ ├── GNM(7,8).pdf │ │ ├── BullGraph.pdf │ │ ├── GNM(6,11).pdf │ │ ├── L(GNM(7,8)).pdf │ │ ├── Subgraphs-G.pdf │ │ ├── 6BlockGraph2.pdf │ │ ├── ShortestPaths.pdf │ │ ├── Subgraphs-H1.pdf │ │ ├── Subgraphs-H2.pdf │ │ ├── Subgraphs-H3.pdf │ │ ├── L(G)Neighbourhood.pdf │ │ ├── 6BlockGraph2-Blocks.pdf │ │ ├── BullGraph.ipe │ │ └── Subgraphs-H1.ipe │ ├── Graph Theory.pdf │ ├── Graph Theory (Small Screen).pdf │ ├── Graph Theory.tex │ ├── Graph Theory (Small Screen).tex │ └── Preamble.tex ├── Testing of Hypotheses │ ├── x + y = log 4.pdf │ ├── RectangularHyperbola.pdf │ └── Testing of Hypotheses.pdf ├── Generating Functions │ └── Generating Functions.pdf ├── MAT 2138 - Discrete Mathematics │ ├── Images │ │ ├── MST.pdf │ │ ├── Path1.pdf │ │ ├── Prim.pdf │ │ ├── Walk1.pdf │ │ ├── Walks.pdf │ │ ├── Cycle1.pdf │ │ ├── Dijkstra.pdf │ │ ├── GNM(6,8).pdf │ │ ├── GNM(7,8).pdf │ │ ├── BullGraph.pdf │ │ ├── GNM(6,11).pdf │ │ ├── 6BlockGraph2.pdf │ │ ├── L(GNM(7,8)).pdf │ │ ├── ShortestPaths.pdf │ │ ├── Subgraphs-G.pdf │ │ ├── Subgraphs-H1.pdf │ │ ├── Subgraphs-H2.pdf │ │ ├── Subgraphs-H3.pdf │ │ ├── L(G)Neighbourhood.pdf │ │ ├── 6BlockGraph2-Blocks.pdf │ │ └── BullGraph.ipe │ ├── MAT 2138 - Discrete Mathematics.pdf │ ├── MAT 2138 - Discrete Mathematics (Small Screen).pdf │ ├── MAT 2138 - Discrete Mathematics.tex │ ├── MAT 2138 - Discrete Mathematics (Small Screen).tex │ └── Preamble.tex ├── Theory of Computation │ ├── Theory of Computation.pdf │ ├── Theory of Computation (Small Screen).pdf │ ├── Theory of Computation (Small Screen).tex │ └── Preamble.tex ├── Applied Linear Algebra │ └── Applied Linear Algebra.pdf ├── Category Theory for Algebra II │ ├── Mono=Injection.pdf │ ├── Category Theory for Algebra II.pdf │ ├── Category Theory for Algebra II (Small Screen).pdf │ ├── References.bib │ ├── Category Theory for Algebra II.tex │ ├── Category Theory for Algebra II (Small Screen).tex │ └── Preamble.tex ├── Estimation of Parameters │ ├── Estimation of Parameters.pdf │ └── Estimation of Parameters.tex ├── Ordering of Permutations │ └── Ordering of Permutations.pdf ├── Computational Mathematics │ ├── Computational Mathematics.pdf │ ├── Computational Mathematics (Small Screen).pdf │ ├── Computational Mathematics (Small Screen).tex │ └── Computational Mathematics.tex └── Introduction to Category Theory │ └── Introduction to Category Theory.pdf ├── Problem Sets ├── Probability and Statistics │ ├── Set 1.pdf │ ├── Set 2.pdf │ ├── Set 3.pdf │ ├── Set 4.pdf │ ├── Set 5.pdf │ ├── Set 6.pdf │ ├── Set 7.pdf │ ├── Set 8.pdf │ ├── Set 9.pdf │ ├── Set 10.pdf │ ├── Set3Graph.pdf │ ├── N(0,1) Table.png │ ├── Chi-Square Table.png │ ├── Statistical Tables.pdf │ ├── Tables of Distributions.pdf │ ├── Set 6.tex │ ├── Set 8.tex │ ├── Set 7.tex │ ├── Set 4.tex │ ├── Set 9.tex │ ├── Set 10.tex │ ├── Set 2.tex │ ├── Set 3.tex │ ├── Set 5.tex │ └── Set 1.tex ├── Lattice Theory │ ├── Lattice Theory Problems.pdf │ └── Lattice Theory Problems.tex ├── MAT 2155 - Combinatorics │ ├── Combinatorics - Set 1.pdf │ ├── Combinatorics - Set 2.pdf │ ├── Combinatorics - Set 3.pdf │ ├── Combinatorics - Set 4.pdf │ ├── Combinatorics - Set 5.pdf │ ├── Combinatorics - Set 3.tex │ ├── Combinatorics - Set 5.tex │ ├── Combinatorics - Set 2.tex │ ├── Combinatorics - Set 1.tex │ └── Combinatorics - Set 4.tex └── MAC 1103 - Computational Mathematics │ ├── Integer Partitions.pdf │ ├── Generating Functions.pdf │ ├── Propositional Calculus.pdf │ ├── Integer Partitions.tex │ ├── Propositional Calculus.tex │ └── Generating Functions.tex ├── Programming └── MAC 2203 │ ├── CountingAndRadixSort.py │ ├── BST.py │ ├── BSTFP.py │ ├── RBTreeFP.py │ └── RBTree.py ├── Notebooks └── Permutations.hs └── .gitignore /Obsidian/.obsidian/app.json: -------------------------------------------------------------------------------- 1 | {} 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-------------------------------------------------------------------------------- /Notes/Applied Graph Theory/Applied Graph Theory.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, a4paper, 12pt]{article} 2 | \usepackage[top = 30mm, bottom = 18mm, left=20mm, right = 20mm]{geometry} 3 | 4 | \input{Preamble} 5 | 6 | \begin{document} 7 | \include{Content} 8 | \end{document} -------------------------------------------------------------------------------- /Notes/Applied Graph Theory/Applied Graph Theory (Small Screen).tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, 12pt]{article} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \input{Preamble} 5 | 6 | \begin{document} 7 | \include{Content} 8 | \end{document} -------------------------------------------------------------------------------- /Notes/Category Theory for Algebra II/References.bib: -------------------------------------------------------------------------------- 1 | @misc{SchalkSimmons2005, 2 | author = {A. Schalk and H. Simmons}, 3 | title = {An introduction to category theory in four easy movements}, 4 | year = {2005}, 5 | note = {Notes for a course offered as part of the MSc in Mathematical Logic, Manchester University} 6 | } 7 | -------------------------------------------------------------------------------- /Notes/MAT 6102 - Graph Theory/Graph Theory.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, a4paper, 12pt]{article} 2 | \usepackage[top = 30mm, bottom = 18mm, left=20mm, right = 20mm]{geometry} 3 | 4 | \input{Preamble} 5 | 6 | \newcommand{\spliteq}{} 7 | \newcommand{\splitalign}{} 8 | 9 | 10 | \begin{document} 11 | \include{Content} 12 | \end{document} -------------------------------------------------------------------------------- /Notes/Category Theory for Algebra II/Category Theory for Algebra II.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, a4paper, 12pt]{article} 2 | \usepackage[top = 30mm, bottom = 18mm, left=20mm, right = 20mm]{geometry} 3 | 4 | \input{Preamble} 5 | 6 | \newcommand{\spliteq}{} 7 | \newcommand{\splitalign}{} 8 | 9 | \begin{document} 10 | \include{Content} 11 | \end{document} -------------------------------------------------------------------------------- /Notes/MAT 2138 - Discrete Mathematics/MAT 2138 - Discrete Mathematics.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, a4paper, 12pt]{article} 2 | \usepackage[top = 30mm, bottom = 18mm, left=20mm, right = 20mm]{geometry} 3 | 4 | \input{Preamble} 5 | 6 | \newcommand{\spliteq}{} 7 | \newcommand{\splitalign}{} 8 | 9 | 10 | \begin{document} 11 | \include{Content} 12 | \end{document} -------------------------------------------------------------------------------- /Notes/MAT 6102 - Graph Theory/Graph Theory (Small Screen).tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, 12pt]{article} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \input{Preamble} 5 | 6 | \newcommand{\spliteq}{\\} 7 | \newcommand{\splitalign}{&} 8 | 9 | \begin{document} 10 | \include{Content} 11 | \end{document} -------------------------------------------------------------------------------- /Notes/Theory of Computation/Theory of Computation (Small Screen).tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, 12pt]{article} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \input{Preamble} 5 | 6 | \newcommand{\spliteq}{\\} 7 | \newcommand{\splitalign}{&} 8 | 9 | \begin{document} 10 | \include{Content} 11 | \end{document} -------------------------------------------------------------------------------- /Notes/Category Theory for Algebra II/Category Theory for Algebra II (Small Screen).tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, 12pt]{article} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \input{Preamble} 5 | 6 | \newcommand{\spliteq}{\\} 7 | \newcommand{\splitalign}{&} 8 | 9 | \begin{document} 10 | \include{Content} 11 | \end{document} -------------------------------------------------------------------------------- /Notes/MAT 2138 - Discrete Mathematics/MAT 2138 - Discrete Mathematics (Small Screen).tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, 12pt]{article} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \input{Preamble} 5 | 6 | \newcommand{\spliteq}{\\} 7 | \newcommand{\splitalign}{&} 8 | 9 | \begin{document} 10 | \include{Content} 11 | \end{document} -------------------------------------------------------------------------------- /Obsidian/.obsidian/core-plugins.json: -------------------------------------------------------------------------------- 1 | [ 2 | "file-explorer", 3 | "global-search", 4 | "switcher", 5 | "graph", 6 | "backlink", 7 | "canvas", 8 | "outgoing-link", 9 | "tag-pane", 10 | "page-preview", 11 | "daily-notes", 12 | "templates", 13 | "note-composer", 14 | "command-palette", 15 | "editor-status", 16 | "bookmarks", 17 | "outline", 18 | "word-count", 19 | "file-recovery" 20 | ] -------------------------------------------------------------------------------- /Obsidian/.obsidian/core-plugins-migration.json: -------------------------------------------------------------------------------- 1 | { 2 | "file-explorer": true, 3 | "global-search": true, 4 | "switcher": true, 5 | "graph": true, 6 | "backlink": true, 7 | "canvas": true, 8 | "outgoing-link": true, 9 | "tag-pane": true, 10 | "properties": false, 11 | "page-preview": true, 12 | "daily-notes": true, 13 | "templates": true, 14 | "note-composer": true, 15 | "command-palette": true, 16 | "slash-command": false, 17 | "editor-status": true, 18 | "bookmarks": true, 19 | "markdown-importer": false, 20 | "zk-prefixer": false, 21 | "random-note": false, 22 | "outline": true, 23 | "word-count": true, 24 | "slides": false, 25 | "audio-recorder": false, 26 | "workspaces": false, 27 | "file-recovery": true, 28 | "publish": false, 29 | "sync": false 30 | } -------------------------------------------------------------------------------- /Programming/MAC 2203/CountingAndRadixSort.py: -------------------------------------------------------------------------------- 1 | import numpy as np 2 | 3 | def countingSort(a): 4 | k = max(map(lambda p: p[0], a)) 5 | # Or: max(a)[0] 6 | b = np.array(a) 7 | c = np.array([0]*(k + 1)) 8 | 9 | for (x,i) in a: 10 | c[x] += 1 11 | 12 | for i in range(1, k + 1): 13 | c[i] = c[i] + c[i - 1] 14 | 15 | for i in range(len(a), 0, -1): 16 | b[c[a[i - 1][0]] - 1] = a[i - 1] 17 | c[a[i - 1][0]] = c[a[i - 1][0]] - 1 18 | 19 | return b 20 | 21 | def _radixSort(a): 22 | n = len(a) # Number of integers to be sorted 23 | d = len(a[0]) # Number of digits in each integer 24 | b = np.array(a) 25 | 26 | for t in range(d - 1, -1, -1): 27 | c = np.array([0]*10) 28 | for i in range(n): 29 | c[a[i][t]] += 1 30 | 31 | for i in range(1, 10): 32 | c[i] = c[i] + c[i - 1] 33 | 34 | for i in range(n, 0, -1): 35 | b[c[a[i - 1][t]] - 1] = a[i - 1] 36 | c[a[i - 1][t]] = c[a[i - 1][t]] - 1 37 | a = np.array(b) 38 | 39 | return map(list, b) 40 | 41 | def intToDigits(n): 42 | if n < 10: 43 | return [n] 44 | else: 45 | return intToDigits(n//10) + [n%10] 46 | 47 | def digitsToInt(ds): 48 | if ds == []: 49 | return 0 50 | else: 51 | return ds[-1] + 10*digitsToInt(ds[:-1]) 52 | 53 | def homogenise(a): 54 | d = max(map(len, a)) 55 | return list(map(lambda ds: (d - len(ds))*[0] + ds, a)) 56 | 57 | def listMap(f, xs): 58 | return list(map(f, xs)) 59 | 60 | def radixSort(a): 61 | return listMap(digitsToInt, _radixSort(homogenise(listMap(intToDigits, a)))) 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | -------------------------------------------------------------------------------- /Problem Sets/MAT 2155 - Combinatorics/Combinatorics - Set 3.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = xelatex 2 | 3 | \documentclass[svgnames]{amsart} 4 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[T1]{fontenc} 12 | \usepackage[sfdefault, lining, scale=0.9]{FiraSans} 13 | \usepackage{kmath} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | \linespread{1.2} 24 | 25 | \title{MAT 2155: Problem Set 3} 26 | \date{} 27 | 28 | \begin{document} 29 | \maketitle 30 | \begin{enumerate}[leftmargin=*] 31 | \item In a class of $76$ students, $26$ have a brother, $29$ have a sister, and $20$ have both. How many have no siblings? 32 | 33 | \item Number of positive integers between $1$ and $100$ not divisible by any of $2$, $3$, and $5$. 34 | 35 | \item Number of positive integers less than or equal to $p^a q^b$, not divisible by either one of $p$ and $q$, where $p$ and $q$ are distinct prime numbers and $a$ and $b$ are positive integers. 36 | 37 | \item Number of derangements of $n$ objects. 38 | 39 | \item Show that the proportion of permutations of $1, 2, \ldots, n$ with no consecutive pair $i$, $i + 1$ for any $i$ is $\frac{n + 1}{n\,e}$. 40 | 41 | \item Number of distributions of $30$ distinct objects into $3$ distinct boxes such that no box is empty. 42 | 43 | \item Number of subsets of $\{1, 2, \ldots, 2n\}$ such that the sum of all the elements in the subset is odd. 44 | 45 | \item If $A$ and $B$ are finite sets of cardinalities $n$ and $m$ respectively, where $n \ge m$, then show that the number of surjective functions from $A$ to $B$ is $\sum\limits_{k=0}^m (-1)^k \binom m k (m - k)^n$. 46 | 47 | \end{enumerate} 48 | \end{document} -------------------------------------------------------------------------------- /Notebooks/Permutations.hs: -------------------------------------------------------------------------------- 1 | module Permutations where 2 | 3 | fact n = foldl (*) 1 [2..n] 4 | 5 | del k xs = [x | x <- xs, x /= k] 6 | 7 | count p xs = length [x | x <- xs, p x] 8 | 9 | --The number of the permutation in lexicographical order. Usage example: lexNumOfPerm [2, 4, 3, 1] 10 | lexNumOfPerm [] = 1 11 | lexNumOfPerm (x:xs) = lexNumOfPerm xs + count (< x) xs * fact (length xs) 12 | 13 | lexPerm' k acc [] = acc 14 | lexPerm' k acc xs = lexPerm' (k `mod` nf) (mark:acc)(del mark xs) where 15 | mark = xs !! (k `div` nf) 16 | nf = fact (length xs - 1) 17 | 18 | --The k-th permutation in lexicographical order. Usage example: lexPerm 12 [1, 2, 3, 4] 19 | lexPerm k = reverse . lexPerm' (k - 1) [] 20 | 21 | --The k-th permutation in lexicographical order. Usage example: revLexPerm 12 [1, 2, 3, 4] 22 | revLexPerm k xs = lexPerm' (k - 1) [] (reverse xs) 23 | 24 | fikePlaceVals n = foldl (\(x:xs) p -> p*x : x : xs) [1] [n, n - 1.. 3] 25 | 26 | dSeq k n = reverse $ tail $ foldl (\(x:xs) p -> (mod x p) : (div x p) : xs) [k - 1] (fikePlaceVals n) 27 | 28 | fikeSeq k n = zipWith (-) [1..n] (dSeq k $ n) 29 | 30 | swap i j list | i == j = list 31 | | i < j = hs ++ (y:xs) ++ (x:ys) 32 | | otherwise = swap j i list where 33 | (hs, zs) = splitAt i list 34 | (x:xs, y:ys) = splitAt (j - i) zs 35 | 36 | --The k-th permutation in Fike's order. Usage example: fikePerm 12 [1, 2, 3, 4] 37 | fikePerm k marks = snd (foldl (\(i, p) j -> (i + 1, swap i j p)) (1, marks) $ fikeSeq k (length marks)) 38 | 39 | --Powerset of a set. Usage example: powerSet [1, 2, 3, 4] 40 | powerSet [] = [[]]; 41 | powerSet (x:xs) = pXs ++ [x:ys | ys <- pXs] where pXs = powerSet xs 42 | 43 | setFromBits _ 0 = []; setFromBits (x:xs) bits = if (bits `mod` 2 == 0) then setFromBits xs (bits `div` 2) else x : setFromBits xs (bits `div` 2) 44 | 45 | --Powerset of a set using setFromBits 46 | powerSet' set = map (setFromBits set) [0.. 2^n -1] where n = length set 47 | 48 | --nCr 49 | comb n r = product [n, n - 1 .. n - r + 1] `div` product [2..r] 50 | -------------------------------------------------------------------------------- /Programming/MAC 2203/BST.py: -------------------------------------------------------------------------------- 1 | class BST: 2 | nil = True 3 | 4 | def isNil(self): 5 | return self.nil 6 | 7 | def __init__(self, val = None): 8 | if val: 9 | self.insert(val) 10 | 11 | def inorder(self): 12 | if not self.isNil(): 13 | return self.left.inorder() + " " + str(self.val) + " " + self.right.inorder() 14 | else: 15 | return "" 16 | 17 | def insert(self, val): 18 | if self.isNil(): 19 | self.nil = False 20 | self.val = val 21 | self.left = BST() 22 | self.right = BST() 23 | self.parent = BST() 24 | 25 | elif val < self.val: 26 | self.left.insert(val) 27 | self.left.parent = self 28 | else: 29 | self.right.insert(val) 30 | self.right.parent = self 31 | 32 | def find(self, val): 33 | if self.isNil(): 34 | raise Exception(str(val) + " does not exist in the tree") 35 | elif val < self.val: 36 | return self.left.find(val) 37 | elif val > self.val: 38 | return self.right.find(val) 39 | else: 40 | return self 41 | 42 | def minimum(self): 43 | if self.isNil(): 44 | raise Exception("Tree is empty") 45 | elif self.left.isNil(): 46 | return self 47 | else: 48 | return self.left.minimum() 49 | 50 | def maximum(self): 51 | if self.isNil(): 52 | raise Exception("Tree is empty") 53 | elif self.right.isNil(): 54 | return self 55 | else: 56 | return self.right.maximum() 57 | 58 | def successor(self, x): 59 | if not x.right.isNil(): 60 | return x.right.minimum() 61 | y = x.parent 62 | while not y.isNil() and x == y.right: 63 | x = y 64 | y = y.parent 65 | return y 66 | 67 | 68 | t = BST() 69 | for x in [10, 35, 20, 50, 90, 4, 100, 10, 90, 3, 300, 20]: 70 | t.insert(x) 71 | 72 | print(t.inorder()) 73 | 74 | print("\nMinimum: ", t.minimum().val) 75 | print("Maximum: ", t.maximum().val, "\n") 76 | 77 | -------------------------------------------------------------------------------- /Problem Sets/MAT 2155 - Combinatorics/Combinatorics - Set 5.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = xelatex 2 | 3 | \documentclass[svgnames]{amsart} 4 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[T1]{fontenc} 12 | \usepackage[sfdefault, lining, scale=0.9]{FiraSans} 13 | \usepackage{kmath} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | \linespread{1.2} 24 | 25 | \title{MAT 2155: Problem Set 5} 26 | \date{} 27 | 28 | \begin{document} 29 | \maketitle 30 | \begin{enumerate}[leftmargin=*] 31 | \item List and enumerate all compositions of $8$ into $3$ parts. Next, enumerate the partitions of $8$ into $3$ parts. 32 | 33 | \item Show that the number of partitions of $n$ in which no integer occurs more than twice as a part is equal to the number of partitions of $n$ into parts not divisible by $3$. 34 | 35 | \item Show that the number of partitions of $n$ in which every part is odd is equal to the number of partitions of $n$ with unequal parts. 36 | 37 | \item List all self-conjugate partitions of $15$. 38 | 39 | \item Show that the number of partitions of $n$ is equal to the number of partitions of $2n$ into exactly $n$ parts. 40 | 41 | \item Show that the number of partitions of $n$ with $k$ parts is equal to the number of partitions of $n$ with largest part $k$. 42 | 43 | \item Show that the number of partitions of $n$ into three parts such that the largest is not larger than the sum of the other two is equal to the number of partitions of $n$ into $2$s, $3$s, and $4$s. 44 | 45 | \item Show that the number of partitions of $2n + k$ into $n + k$ parts is independent of $k$. 46 | 47 | \item Show that the number of partitions of $n$ in which odd parts are not repeated equals the number of partitions of $n$ in which every part is either odd or a multiple of $4$. 48 | 49 | \item Show that the number of partitions of $n$ with $k$ parts and largest part $m$ is equal to the number of partitions of $n - k$ with $m - 1$ parts, none of which is greater than $k$. 50 | 51 | \end{enumerate} 52 | \end{document} -------------------------------------------------------------------------------- /Notes/Computational Mathematics/Computational Mathematics (Small Screen).tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, 12pt]{article} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{natbib} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | \usepackage[all, cmtip, 2cell]{xy} 8 | \setcounter{tocdepth}{3} 9 | \usepackage{graphicx} 10 | \usepackage{physics} 11 | 12 | \usepackage{tikz} 13 | 14 | \usepackage{mathtools} 15 | \usepackage{xspace, cancel} 16 | 17 | \usepackage{enumitem} 18 | \setlist{noitemsep} 19 | 20 | \usepackage{scrlayer-scrpage} 21 | \ohead{\color{blue!35!black} \scshape VM} 22 | \cfoot*{\pagemark} 23 | 24 | \usepackage{tocloft} 25 | \renewcommand{\cftdot}{} 26 | 27 | \usepackage{hyperref} 28 | \definecolor{linkcolor}{RGB}{32, 96, 192} 29 | \hypersetup{colorlinks, linkcolor = linkcolor, urlcolor = linkcolor, linktocpage = true} 30 | \usepackage{bookmark} 31 | \bookmarksetup{color = [RGB]{32, 96, 192}} 32 | \usepackage[capitalise]{cleveref} 33 | 34 | \usepackage[intoc]{nomencl} 35 | \makenomenclature 36 | 37 | \usepackage[toc, page]{appendix} 38 | 39 | \usepackage{newpxmath} 40 | \usepackage{charter} 41 | \usepackage[T1]{fontenc} 42 | 43 | \newtheorem{Theorem}{Theorem}[section] 44 | \newtheorem{Lemma}[Theorem]{Lemma} 45 | \newtheorem{Corollary}[Theorem]{Corollary} 46 | 47 | \theoremstyle{definition} 48 | \newtheorem{Definition}[Theorem]{Definition} 49 | \newtheorem*{Definition*}{Definition} 50 | \newtheorem{Example}[Theorem]{Example} 51 | \newtheorem*{Example*}{Example} 52 | \newtheorem{Exercise}{Exercise}[section] 53 | 54 | \theoremstyle{remark} 55 | \newtheorem*{Remark*}{Remark} 56 | \newtheorem*{Solution*}{Solution} 57 | \newtheorem*{Note*}{Note} 58 | 59 | 60 | \definecolor{alertcolor}{rgb}{0.8, 0.5, 0} 61 | \newcommand{\newterm}[1]{{\color{alertcolor} #1}} 62 | 63 | \definecolor{hintcolor}{rgb}{0, 0, 0.5} 64 | \newcommand{\hint}[1]{\tiny\color{hintcolor} Hint: #1} 65 | 66 | \DeclareMathOperator{\ord}{o} 67 | \newcommand{\Mod}[1]{\ (\mathrm{mod}\ #1)} 68 | \DeclareMathOperator{\lcm}{lcm} 69 | 70 | \DeclareMathOperator{\im}{im} 71 | \DeclareMathOperator{\dom}{dom} 72 | \DeclareMathOperator{\cod}{cod} 73 | \newcommand{\id}{\mathrm{id}} 74 | \newcommand{\symdiff}{\mathbin{\triangle}} 75 | \newcommand{\nth}{\textsuperscript{th}\xspace} 76 | 77 | 78 | \begin{document} 79 | \include{Content} 80 | \end{document} -------------------------------------------------------------------------------- /Notes/Computational Mathematics/Computational Mathematics.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, a4paper, 12pt]{article} 2 | \usepackage[top = 30mm, bottom = 18mm, left=20mm, right = 20mm]{geometry} 3 | 4 | \usepackage{natbib} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | \usepackage[all, cmtip, 2cell]{xy} 8 | \setcounter{tocdepth}{3} 9 | \usepackage{graphicx} 10 | \usepackage{physics} 11 | 12 | \usepackage{tikz} 13 | 14 | \usepackage{mathtools} 15 | \usepackage{xspace, cancel} 16 | 17 | \usepackage{enumitem} 18 | \setlist{noitemsep} 19 | 20 | \usepackage{scrlayer-scrpage} 21 | \ohead{\color{blue!35!black} \scshape VM} 22 | \cfoot*{\pagemark} 23 | 24 | \usepackage{tocloft} 25 | \renewcommand{\cftdot}{} 26 | 27 | \usepackage{hyperref} 28 | \definecolor{linkcolor}{RGB}{32, 96, 192} 29 | \hypersetup{colorlinks, linkcolor = linkcolor, urlcolor = linkcolor, linktocpage = true} 30 | \usepackage{bookmark} 31 | \bookmarksetup{color = [RGB]{32, 96, 192}} 32 | \usepackage[capitalise]{cleveref} 33 | 34 | \usepackage[intoc]{nomencl} 35 | \makenomenclature 36 | 37 | \usepackage[toc, page]{appendix} 38 | 39 | \usepackage{newpxmath} 40 | \usepackage{charter} 41 | \usepackage[T1]{fontenc} 42 | 43 | \newtheorem{Theorem}{Theorem}[section] 44 | \newtheorem{Lemma}[Theorem]{Lemma} 45 | \newtheorem{Corollary}[Theorem]{Corollary} 46 | 47 | \theoremstyle{definition} 48 | \newtheorem{Definition}[Theorem]{Definition} 49 | \newtheorem*{Definition*}{Definition} 50 | \newtheorem{Example}[Theorem]{Example} 51 | \newtheorem*{Example*}{Example} 52 | \newtheorem{Exercise}{Exercise}[section] 53 | 54 | 55 | \theoremstyle{remark} 56 | \newtheorem*{Remark*}{Remark} 57 | \newtheorem*{Solution*}{Solution} 58 | \newtheorem*{Note*}{Note} 59 | 60 | \crefname{Exercise}{Exercise}{Exercises} 61 | \crefname{Section}{Section}{Sections} 62 | 63 | \definecolor{alertcolor}{rgb}{0.8, 0.5, 0} 64 | \newcommand{\newterm}[1]{{\color{alertcolor} #1}} 65 | 66 | \definecolor{hintcolor}{rgb}{0, 0, 0.5} 67 | \newcommand{\hint}[1]{\tiny\color{hintcolor} Hint: #1} 68 | 69 | \DeclareMathOperator{\ord}{o} 70 | \newcommand{\Mod}[1]{\ (\mathrm{mod}\ #1)} 71 | \DeclareMathOperator{\lcm}{lcm} 72 | 73 | \DeclareMathOperator{\im}{im} 74 | \DeclareMathOperator{\dom}{dom} 75 | \DeclareMathOperator{\cod}{cod} 76 | \newcommand{\id}{\mathrm{id}} 77 | \newcommand{\symdiff}{\mathbin{\triangle}} 78 | \newcommand{\nth}{\textsuperscript{th}\xspace} 79 | 80 | 81 | \begin{document} 82 | \include{Content} 83 | \end{document} -------------------------------------------------------------------------------- /Programming/MAC 2203/BSTFP.py: -------------------------------------------------------------------------------- 1 | from dataclasses import dataclass 2 | 3 | class BST: 4 | isNil: bool 5 | 6 | class Nil(BST): 7 | isNil = True 8 | 9 | def __str__(self): 10 | return "" 11 | 12 | @dataclass 13 | class Node(BST): 14 | val: int 15 | left: BST 16 | right: BST 17 | 18 | def __str__(self): 19 | return "(" + str(self.left) + ")<-" + str(self.val) + "->(" + str(self.right) + ")" 20 | 21 | def isInTree(tree: BST, x: int): 22 | match tree: 23 | case Nil(): 24 | return False 25 | case Node(val, left, right): 26 | if x == val: 27 | return True 28 | elif x < val: 29 | return isInTree(left, x) 30 | else: 31 | return isInTree(right, x) 32 | 33 | def inorder(tree): 34 | match tree: 35 | case Node(val, left, right): 36 | inorder(left) 37 | print(val) 38 | inorder(right) 39 | 40 | def insert(tree, x): 41 | match tree: 42 | case Nil(): 43 | return Node(x, Nil(), Nil()) 44 | case Node(val, left, right): 45 | if x < val: 46 | return Node(val, insert(left, x), right) 47 | else: 48 | return Node(val, left, insert(right, x)) 49 | 50 | def minTree(tree): 51 | match tree: 52 | case Node(_, Nil(), _): 53 | return tree 54 | case Node(_, left, _): 55 | return minTree(left) 56 | 57 | def delete(tree, x): 58 | match tree: 59 | case Nil(): 60 | return tree 61 | case Node(val, left, right): 62 | if x < val: 63 | return Node(val, delete(left, x), right) 64 | elif x > val: 65 | return Node(val, left, delete(right, x)) 66 | else: 67 | match (left, right): 68 | case (Nil(), _): 69 | return right 70 | case (_, Nil()): 71 | return left 72 | case _: 73 | rmin = minTree(right) 74 | return Node(rmin.val, left, delete(right, rmin.val)) 75 | 76 | 77 | t = Nil() 78 | for x in [60, 20, 50, 100, 30, 75, 3, 10, 27]: 79 | t = insert(t, x) 80 | 81 | print(t) 82 | 83 | for x in [3, 100, 27, 60, 20, 75, 10, 50, 30]: 84 | t = delete(t, x) 85 | print(t) 86 | -------------------------------------------------------------------------------- /Problem Sets/MAC 1103 - Computational Mathematics/Integer Partitions.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = xelatex 2 | 3 | \documentclass[svgnames]{amsart} 4 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[T1]{fontenc} 12 | \usepackage[sfdefault, lining, scale=0.9]{FiraSans} 13 | \usepackage{kmath} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | \linespread{1.2} 24 | 25 | \title{MAC 1103: Integer Partitions} 26 | \date{} 27 | 28 | \begin{document} 29 | \maketitle 30 | \begin{enumerate}[leftmargin=*] 31 | \item List and enumerate all compositions of $8$ into $3$ parts. Next, enumerate the partitions of $8$ into $3$ parts. 32 | 33 | \item Show that the number of partitions of $n$ in which every part is odd is equal to the number of partitions of $n$ with unequal parts. 34 | 35 | \item Show that the number of partitions of $n$ in which no integer occurs more than twice as a part is equal to the number of partitions of $n$ into parts not divisible by $3$. 36 | 37 | \item List all self-conjugate partitions of $15$. 38 | 39 | \item Show that the number of partitions of $n$ is equal to the number of partitions of $2n$ into exactly $n$ parts. 40 | 41 | \item Show that the number of partitions of $n$ with $k$ parts is equal to the number of partitions of $n$ with largest part $k$. 42 | 43 | \item Show that the number of partitions of $n$ into three parts such that the largest is not larger than the sum of the other two is equal to the number of partitions of $n$ into $2$s, $3$s, and $4$s. 44 | 45 | \item Show that the number of partitions of $2n + k$ into $n + k$ parts is independent of $k$. 46 | 47 | \item Show that the number of self-conjugate partitions of $n$ is equal to the number of partitions of $n$ into distinct odd parts. 48 | 49 | \item Show that the number of partitions of $n$ in which odd parts are not repeated equals the number of partitions of $n$ in which every part is either odd or a multiple of $4$. 50 | 51 | \item Show that the number of partitions of $n$ with $k$ parts and largest part $m$ is equal to the number of partitions of $n - k$ with $m - 1$ parts, none of which is greater than $k$. 52 | 53 | \end{enumerate} 54 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/MAC 1103 - Computational Mathematics/Propositional Calculus.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = xelatex 2 | 3 | \documentclass[svgnames]{amsart} 4 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm, physics} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[T1]{fontenc} 12 | \usepackage[sfdefault, lining, scale=0.9]{FiraSans} 13 | \usepackage{kmath} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | \linespread{1.2} 24 | 25 | \title{MAC 1103: Propositional Calculus} 26 | \date{} 27 | 28 | \begin{document} 29 | \maketitle 30 | \begin{enumerate}[leftmargin=*] 31 | \item Prove that $S \vee R$ follows from the premises $P \vee Q$, $P \to R$, and $Q \to S$. 32 | 33 | \item Show that $(P \vee Q) \wedge (Q \to R) \wedge (P \to M) \wedge (\neg M) \implies R \wedge (P \vee Q)$. 34 | 35 | \item Show that the premises $P \to (Q \to S)$, $\neg R \vee P$, and $Q$ tautologically imply $R \to S$. 36 | 37 | \item Show that the following sets of premises are (each) inconsistent: 38 | \begin{enumerate} 39 | \item $P \to Q$, $Q \to R$, $R \to \neg P$. 40 | \item $A \vee B$, $A \to \neg C$, $C \to \neg B$, $C$. 41 | \item $P \to Q$, $Q \to R$, $Q \to \neg R$, $P$. 42 | \item $A \to (B \to C)$, $D \to (B \wedge \neg C)$, $A \wedge D$. 43 | \end{enumerate} 44 | 45 | \item Show that the following premises are inconsistent: 46 | \begin{enumerate} 47 | \item If Jack misses many classes due to illness, then he fails school. 48 | \item If Jack fails school, then he is uneducated. 49 | \item If Jack reads a lot of books, then he is not uneducated. 50 | \item Jack misses many classes due to illness and reads a lot of books. 51 | \end{enumerate} 52 | 53 | \item Let the propositions $P$, $Q$, and $R$ be defined as follows: \\ 54 | $P$: $\sqrt 2$ is irrational. \\ 55 | $Q$: $\pqty{\sqrt 2}^{\sqrt 2}$ is rational. \\ 56 | $R$: $\pqty{ \pqty{\sqrt 2}^{\sqrt 2} }^{\sqrt 2} = 2$ is rational. \\ 57 | $S$: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. 58 | 59 | Prove that $S$ follows from the premises $P$, $R$, $P \wedge Q \to S$, and $P \wedge (\neg Q) \wedge R \to S$. That is, show that there exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. 60 | 61 | \item 62 | \end{enumerate} 63 | \end{document} -------------------------------------------------------------------------------- /Notes/MAT 6102 - Graph Theory/Preamble.tex: -------------------------------------------------------------------------------- 1 | % !TeX root = Graph Theory.tex 2 | 3 | \usepackage{natbib} 4 | 5 | \usepackage{amsmath, amsfonts, amssymb, amsthm, bbm} 6 | \usepackage[all, cmtip, 2cell]{xy} 7 | \setcounter{tocdepth}{3} 8 | \usepackage{graphicx, subcaption} 9 | \usepackage{physics} 10 | 11 | \usepackage{tikz} 12 | 13 | \usepackage{mathtools} 14 | \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} 15 | \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} 16 | 17 | \usepackage{xspace, cancel} 18 | 19 | \usepackage{enumitem, array} 20 | \setlist{noitemsep} 21 | 22 | \usepackage{scrlayer-scrpage} 23 | \ohead{\color{blue!35!black} \scshape VM} 24 | \cfoot*{\pagemark} 25 | 26 | \usepackage{tocloft} 27 | \renewcommand{\cftdot}{} 28 | 29 | \usepackage{hyperref} 30 | \definecolor{linkcolor}{RGB}{32, 96, 192} 31 | \hypersetup{colorlinks, linkcolor = linkcolor, urlcolor = linkcolor, linktocpage = true} 32 | \usepackage{bookmark} 33 | \bookmarksetup{color = [RGB]{32, 96, 192}} 34 | 35 | \usepackage[capitalise]{cleveref} 36 | \crefformat{enumi}{#2\textup{#1}#3} 37 | 38 | \usepackage{algorithm, algpseudocode} 39 | 40 | \usepackage{kbordermatrix} 41 | 42 | \usepackage[intoc]{nomencl} 43 | \makenomenclature 44 | 45 | \usepackage[toc, page]{appendix} 46 | 47 | \usepackage{newpxmath} 48 | \usepackage{charter} 49 | \usepackage[T1]{fontenc} 50 | 51 | \newtheorem{Theorem}{Theorem}[section] 52 | \newtheorem{Lemma}[Theorem]{Lemma} 53 | \newtheorem{Corollary}[Theorem]{Corollary} 54 | \newtheorem{Observation}[Theorem]{Observation} 55 | 56 | \theoremstyle{definition} 57 | \newtheorem{Definition}[Theorem]{Definition} 58 | \newtheorem*{Definition*}{Definition} 59 | \newtheorem{Example}[Theorem]{Example} 60 | \newtheorem*{Example*}{Example} 61 | \newtheorem{Exercise}{Exercise}[section] 62 | 63 | \theoremstyle{remark} 64 | \newtheorem*{Remark*}{Remark} 65 | \newtheorem*{Solution*}{Solution} 66 | \newtheorem*{Note*}{Note} 67 | 68 | 69 | \definecolor{alertcolor}{rgb}{0.8, 0.5, 0} 70 | \newcommand{\newterm}[1]{{\color{alertcolor} #1}} 71 | 72 | \definecolor{hintcolor}{rgb}{0, 0, 0.5} 73 | \newcommand{\hint}[1]{{\tiny\color{hintcolor}\textbf{Hint:} #1}} 74 | 75 | \DeclareMathOperator{\Nb}{N} 76 | \DeclareMathOperator{\lcm}{lcm} 77 | \DeclareMathOperator{\im}{im} 78 | \DeclareMathOperator{\dom}{dom} 79 | \DeclareMathOperator{\cod}{cod} 80 | \DeclareMathOperator{\ecc}{ecc} 81 | \DeclareMathOperator{\rad}{rad} 82 | \DeclareMathOperator{\diam}{diam} 83 | \DeclareMathOperator{\LG}{\mathcal L} 84 | \newcommand{\id}{\mathrm{id}} 85 | \newcommand{\symdiff}{\mathbin{\triangle}} 86 | \newcommand{\nth}{\textsuperscript{th}\xspace} -------------------------------------------------------------------------------- /Notes/Applied Graph Theory/Preamble.tex: -------------------------------------------------------------------------------- 1 | % !TeX root = Applied Graph Theory.tex 2 | 3 | \usepackage{natbib} 4 | 5 | \usepackage{amsmath, amsfonts, amssymb, amsthm, bbm} 6 | \usepackage[all, cmtip, 2cell]{xy} 7 | \setcounter{tocdepth}{3} 8 | \usepackage{graphicx, subcaption} 9 | \usepackage{physics} 10 | 11 | \usepackage{tikz} 12 | 13 | \usepackage{mathtools} 14 | \usepackage{xspace, cancel} 15 | 16 | \usepackage{enumitem, array} 17 | \setlist{noitemsep} 18 | 19 | \usepackage{scrlayer-scrpage} 20 | \ohead{\color{blue!35!black} \scshape VM} 21 | \cfoot*{\pagemark} 22 | 23 | \usepackage{tocloft} 24 | \renewcommand{\cftdot}{} 25 | 26 | \usepackage{hyperref} 27 | \definecolor{linkcolor}{RGB}{32, 96, 192} 28 | \hypersetup{colorlinks, linkcolor = linkcolor, urlcolor = linkcolor, linktocpage = true} 29 | \usepackage{bookmark} 30 | \bookmarksetup{color = [RGB]{32, 96, 192}} 31 | 32 | \usepackage[capitalise]{cleveref} 33 | \crefformat{enumi}{#2\textup{#1}#3} 34 | 35 | \usepackage{algorithm, algpseudocode} 36 | 37 | \usepackage{kbordermatrix} 38 | 39 | \usepackage[intoc]{nomencl} 40 | \makenomenclature 41 | 42 | \usepackage[toc, page]{appendix} 43 | 44 | \usepackage{newpxmath} 45 | \usepackage{charter} 46 | \usepackage[T1]{fontenc} 47 | 48 | \newtheorem{Theorem}{Theorem}[section] 49 | \newtheorem{Lemma}[Theorem]{Lemma} 50 | \newtheorem{Corollary}[Theorem]{Corollary} 51 | \newtheorem{Observation}[Theorem]{Observation} 52 | 53 | \theoremstyle{definition} 54 | \newtheorem{Definition}[Theorem]{Definition} 55 | \newtheorem*{Definition*}{Definition} 56 | \newtheorem{Example}[Theorem]{Example} 57 | \newtheorem*{Example*}{Example} 58 | \newtheorem{Exercise}{Exercise}[section] 59 | 60 | \theoremstyle{remark} 61 | \newtheorem*{Remark*}{Remark} 62 | \newtheorem*{Solution*}{Solution} 63 | \newtheorem*{Note*}{Note} 64 | 65 | 66 | \definecolor{alertcolor}{rgb}{0.8, 0.5, 0} 67 | \newcommand{\newterm}[1]{{\color{alertcolor} #1}} 68 | 69 | \definecolor{hintcolor}{rgb}{0, 0, 0.5} 70 | \newcommand{\hint}[1]{{\tiny\color{hintcolor}\textbf{Hint:} #1}} 71 | 72 | \DeclareMathOperator{\ord}{o} 73 | \newcommand{\Mod}[1]{\ (\mathrm{mod}\ #1)} 74 | \DeclareMathOperator{\lcm}{lcm} 75 | 76 | \DeclareMathOperator{\im}{im} 77 | \DeclareMathOperator{\dom}{dom} 78 | \DeclareMathOperator{\cod}{cod} 79 | \DeclareMathOperator{\ecc}{ecc} 80 | \DeclareMathOperator{\rad}{rad} 81 | \DeclareMathOperator{\diam}{diam} 82 | \newcommand{\id}{\mathrm{id}} 83 | \newcommand{\symdiff}{\mathbin{\triangle}} 84 | \newcommand{\nth}{\textsuperscript{th}\xspace} 85 | \DeclareMathOperator{\bestDTo}{bestDTo} 86 | \DeclareMathOperator{\tree}{tree} 87 | 88 | \algrenewcommand\algorithmicrequire{\textbf{Input:}} -------------------------------------------------------------------------------- /Programming/MAC 2203/RBTreeFP.py: -------------------------------------------------------------------------------- 1 | from dataclasses import dataclass 2 | from enum import Enum 3 | 4 | class Color(Enum): 5 | B = 0 6 | R = 1 7 | 8 | def colStr(col: Color): 9 | return "B" if col == Color.B else "R" 10 | 11 | class Tree: 12 | col: Color 13 | 14 | class E: 15 | col = Color.B 16 | 17 | def __repr__(self): 18 | return "" 19 | 20 | @dataclass 21 | class T: 22 | col: Color 23 | left: Tree 24 | val: int 25 | right: Tree 26 | 27 | def __repr__(self): 28 | return "(" + str(self.left) + ")<-" + colStr(self.col) + str(self.val) + "->(" + str(self.right) + ")" 29 | 30 | def isEmpty(t): 31 | match t: 32 | case E(): 33 | return True 34 | case _: 35 | return False 36 | 37 | def isIn(t, x): 38 | match t: 39 | case E(): 40 | return False 41 | case T(_, left, val, right): 42 | if x == val: 43 | return True 44 | elif x < val: 45 | return isIn(left, x) 46 | else: 47 | return isIn(right, x) 48 | 49 | def makeBlack(t): 50 | match t: 51 | case T(_, left, val, right): 52 | return T(Color.B, left, val, right) 53 | 54 | def balance(col, t1, z, t2): 55 | match (col, t1, z, t2): 56 | case (Color.B, T(Color.R, T(Color.R, a, x, b), y, c), z, d): 57 | return T(Color.R, T(Color.B, a, x, b), y, T(Color.B, c, z, d)) 58 | case (Color.B, T(Color.R, a, x, T(Color.R, b, y, c)), z, d): 59 | return T(Color.R, T(Color.B, a, x, b), y, T(Color.B, c, z, d)) 60 | case (Color.B, a, x, T(Color.R, b, y, T(Color.R, c, z, d))): 61 | return T(Color.R, T(Color.B, a, x, b), y, T(Color.B, c, z, d)) 62 | case (Color.B, a, x, T(Color.R, T(Color.R, b, y, c), z, d)): 63 | return T(Color.R, T(Color.B, a, x, b), y, T(Color.B, c, z, d)) 64 | case _: 65 | return T(col, t1, z, t2) 66 | 67 | def ins(t, x): 68 | match t: 69 | case E(): 70 | return T(Color.R, E(), x, E()) 71 | case T(col, l, y, r): 72 | if x < y: 73 | return balance(col, ins(l, x), y, r) 74 | elif x > y: 75 | return balance(col, l, y, ins(r, x)) 76 | else: 77 | return t 78 | 79 | def insert(t, x): 80 | return makeBlack(ins(t, x)) 81 | 82 | 83 | t = E() 84 | 85 | for x in [60, 20, 50, 100, 30, 75, 3, 10, 27]: 86 | t = insert(t, x) 87 | 88 | print(t) 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | -------------------------------------------------------------------------------- /Problem Sets/Lattice Theory/Lattice Theory Problems.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = xelatex 2 | 3 | \documentclass[svgnames]{amsart} 4 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | 8 | \usepackage{physics} 9 | 10 | \usepackage{enumitem} 11 | \setlist[enumerate,1]{label=\arabic*.} 12 | 13 | \usepackage[T1]{fontenc} 14 | \usepackage[sfdefault, lining, scale=0.9]{FiraSans} 15 | \usepackage{kmath} 16 | 17 | \usepackage{xcolor} 18 | \usepackage{scrlayer-scrpage} 19 | \ohead{\color{blue!35!black} \scshape VM} 20 | \cfoot*{\pagemark} 21 | 22 | \renewcommand{\th}{\textsuperscript{th}} 23 | 24 | \setlength{\parindent}{0pt} 25 | \linespread{1.2} 26 | 27 | \title{Lattice Theory Problems} 28 | \date{} 29 | 30 | \begin{document} 31 | \maketitle 32 | \begin{enumerate}[leftmargin=*] 33 | \item If $x$ and $y$ are two elements of a lattice, show that $x \wedge y = y$ if and only if $x \vee y = x$. 34 | 35 | \item If $x$, $y$, and $z$ are elements of a lattice, show that 36 | \begin{align*} 37 | x \vee (y \wedge z) \le (x \vee y) \wedge (x \vee z) \\ 38 | (x \wedge y) \vee (x \wedge z) \le x \wedge (y \vee z). 39 | \end{align*} 40 | 41 | \item Show that in an algebraic system $(L, \oplus, \otimes)$, where $\oplus$ and $\otimes$ are binary operations satisfying the absorption law, $\oplus$ and $\otimes$ are idempotent. 42 | 43 | \item Let $a$, $b$, $c$ be elements in a lattice $(L, \le)$. Show that $a \le b$ if and only if 44 | \begin{equation*} 45 | a \vee (b \wedge c) \le b \wedge (a \vee c). 46 | \end{equation*} 47 | 48 | \item Show that a lattice $L$ is distributive if and only if for all elements $x, y, z \in L$, $(x \vee y) \wedge z \le x \vee (y \wedge z)$. 49 | 50 | \item Show that every chain is a distributive lattice. Which chains are Boolean lattices? 51 | 52 | \item Let $L$ be a distributive lattice. Show that for $a, b \in L$, if there exists an element $x \in L$ such that $a \vee x = b \vee x$ and $a \wedge x = b \wedge x$, then $a = b$. 53 | 54 | \item Give an example of a complemented lattice that is not distributive. 55 | 56 | \item Does the lattice $(\mathbb N, \mid)$ (where $\mathbb N = \{1, 2, 3, \ldots\}$) contain 57 | \begin{enumerate} 58 | \item a universal lower bound? 59 | \item a universal upper bound? 60 | \end{enumerate} 61 | 62 | \item Does the lattice $(\mathbb N_0, \mid)$ (where $\mathbb N_0 = \mathbb N \cup \{0\}$) contain a universal upper bound? 63 | 64 | \item Show that every finite lattice contains a universal upper bound and a universal lower bound. 65 | 66 | \item Show that if a lattice contains both $0$ and $1$, then they are the unique complements of each other. 67 | 68 | \item Compute the CNF and DNF of the expression $E(a, b, c) = \overline{\pqty{ a \wedge \pqty{ \overline b \vee \pqty{ \overline c \wedge a } } }}$ over the $2$-valued Boolean algebra. 69 | \end{enumerate} 70 | \end{document} -------------------------------------------------------------------------------- /Notes/Category Theory for Algebra II/Preamble.tex: -------------------------------------------------------------------------------- 1 | % !TeX root = Category Theory for Algebra II.tex 2 | 3 | \usepackage{amsmath, amsfonts, amssymb, amsthm, bbm} 4 | \usepackage[all, cmtip, 2cell]{xy} 5 | \setcounter{tocdepth}{3} 6 | \usepackage{graphicx} 7 | \usepackage[subrefformat=parens,labelformat=parens]{subcaption} 8 | \usepackage{physics} 9 | 10 | \usepackage{tikz} 11 | 12 | \usepackage{mathtools} 13 | \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} 14 | \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} 15 | 16 | \usepackage{xspace, cancel} 17 | 18 | \usepackage{enumitem, array} 19 | \setlist{noitemsep} 20 | 21 | \usepackage{scrlayer-scrpage} 22 | \ohead{\color{blue!35!black} \scshape VM} 23 | \cfoot*{\pagemark} 24 | 25 | \usepackage{tocloft} 26 | \renewcommand{\cftdot}{} 27 | 28 | \usepackage{xcolor} 29 | \usepackage{hyperref} 30 | \definecolor{linkcolor}{RGB}{32, 96, 192} 31 | \hypersetup{colorlinks, linkcolor = linkcolor, urlcolor = linkcolor, citecolor = red, linktocpage = true} 32 | \usepackage{bookmark} 33 | \bookmarksetup{color = [RGB]{32, 96, 192}} 34 | 35 | \usepackage[capitalise]{cleveref} 36 | \crefformat{enumi}{#2\textup{#1}#3} 37 | 38 | \usepackage{algorithm, algpseudocode} 39 | 40 | \usepackage{kbordermatrix} 41 | 42 | \usepackage[intoc]{nomencl} 43 | \makenomenclature 44 | 45 | \usepackage[toc, page]{appendix} 46 | 47 | \usepackage{newpxmath} 48 | \usepackage{charter} 49 | \usepackage[T1]{fontenc} 50 | 51 | \usepackage{MnSymbol} 52 | 53 | \newtheorem{Theorem}{Theorem}[section] 54 | \newtheorem{Lemma}[Theorem]{Lemma} 55 | \newtheorem{Corollary}[Theorem]{Corollary} 56 | \newtheorem{Observation}[Theorem]{Observation} 57 | 58 | \theoremstyle{definition} 59 | \newtheorem{Definition}[Theorem]{Definition} 60 | \newtheorem*{Definition*}{Definition} 61 | \newtheorem{Example}[Theorem]{Example} 62 | \newtheorem*{Example*}{Example} 63 | \newtheorem{Exercise}{Exercise}[section] 64 | 65 | \theoremstyle{remark} 66 | \newtheorem*{Remark*}{Remark} 67 | \newtheorem*{Solution*}{Solution} 68 | 69 | 70 | \definecolor{alertcolor}{rgb}{0.8, 0.5, 0} 71 | \newcommand{\newterm}[1]{{\color{alertcolor} #1}} 72 | 73 | \definecolor{hintcolor}{rgb}{0, 0, 0.5} 74 | \newcommand{\hint}[1]{{\noindent\tiny\color{hintcolor}\textbf{Hint:} #1}} 75 | 76 | \definecolor{notecolor}{rgb}{0, 0.5, 0.1} 77 | \newcommand{\note}[1]{{\noindent\footnotesize\color{notecolor}\textbf{Note.} #1}} 78 | 79 | \newcommand{\cat}[1]{\mathsf{#1}} 80 | \newcommand{\opp}{^\mathsf{op}} 81 | \DeclareMathOperator{\Ob}{Ob} 82 | \DeclareMathOperator{\Ar}{Ar} 83 | \DeclareMathOperator{\Hom}{Hom} 84 | \DeclareMathOperator{\Mor}{Mor} 85 | \newcommand{\fun}[1]{\mathcal{#1}} 86 | \DeclareMathOperator{\lcm}{lcm} 87 | \DeclareMathOperator{\im}{im} 88 | \DeclareMathOperator{\dom}{dom} 89 | \DeclareMathOperator{\cod}{cod} 90 | \let\Re\relax 91 | \DeclareMathOperator{\Re}{Re} 92 | \let\Im\relax 93 | \DeclareMathOperator{\Im}{Im} 94 | \newcommand{\id}{\mathrm{id}} 95 | \newcommand{\nth}{\textsuperscript{th}\xspace} -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 6.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx, tikz} 6 | \usepackage{mathtools, physics} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[default,bold]{sourceserifpro} 12 | \usepackage[T1]{fontenc} 13 | \usepackage[]{fouriernc} 14 | 15 | \usepackage{xcolor} 16 | 17 | \renewcommand{\th}{\textsuperscript{th}} 18 | 19 | \setlength{\parindent}{0pt} 20 | 21 | \title[]{Probability and Statistics -- Problem Set 6} 22 | 23 | \DeclareMathOperator{\Prob}{P} 24 | \DeclareMathOperator{\EV}{\mathbb E} 25 | \DeclareMathOperator{\Var}{\mathbb V} 26 | 27 | 28 | \begin{document} 29 | \maketitle 30 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 31 | \item If $Z \sim N(0,1)$, compute 32 | \begin{enumerate} 33 | \item $\Prob[Z < 1]$ 34 | \item $\Prob[Z < 2.4]$ 35 | \item $\Prob[1 < Z < 2.4]$ 36 | \item $\Prob[Z < -1]$ 37 | \item $\Prob[-2.4 < Z < 1]$. 38 | \item $\Prob[|Z| > 1]$ 39 | \end{enumerate} 40 | 41 | \item If $X \sim N(175, 25^2)$, compute 42 | \begin{enumerate} 43 | \item $\Prob[X < 200]$ 44 | \item $\Prob[X < 235]$ 45 | \item $\Prob[200 < X < 235]$ 46 | \item $\Prob[X < 150]$ 47 | \item $\Prob[115 < X < 200]$. 48 | \end{enumerate} 49 | 50 | \item In a normal distribution, $31\%$ of the observations are under $45$ and $8\%$ are over $64$. Find the mean and standard deviation. 51 | 52 | \item In a normal distribution, $7\%$ of the observations are under $35$ and $89\%$ are under $63$. Find the mean and standard deviation. 53 | 54 | \item Suppose that the life lengths of two electronic devices $D_1$ and $D_2$ (in hours) have distributions $N(40, 36)$ and $N(45, 9)$ respectively. If the device is to be used for a $48$ hour period, which device is to be preferred? 55 | 56 | \item Suppose that the scores of an examination are normally distributed with mean $76$ and standard deviation $15$. The top $15\%$ scores receive grade A and the bottom $10\%$ receive grade F. Find 57 | \begin{enumerate} 58 | \item the minimum score to receive an A 59 | \item the minimum score to pass. 60 | \end{enumerate} 61 | 62 | \item Suppose that the heights of $800$ students are normally distributed with mean $66''$ and standard deviation $5''$. Find the number of students with heights 63 | \begin{enumerate} 64 | \item between $65''$ and $70''$ 65 | \item greater than or equal to $6'$. 66 | \end{enumerate} 67 | 68 | \item In a certain examination, the percentages of candidates passing and getting distinction were $45$ and $9$ respectively. Assuming that the marks are normally distributed, determine the average marks obtained by a candidate if the minimum marks for passing and distinction are $40$ and $75$ respectively. 69 | \end{enumerate} 70 | 71 | \includegraphics[trim = 65 30 50 95, clip, page=2, width=\linewidth]{Statistical Tables.pdf} 72 | 73 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 8.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx, tikz} 6 | \usepackage{mathtools, physics} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[default,bold]{sourceserifpro} 12 | \usepackage[T1]{fontenc} 13 | \usepackage[]{fouriernc} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\scriptsize\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | 24 | \title[]{Probability and Statistics -- Problem Set 8} 25 | 26 | \DeclareMathOperator{\Prob}{P} 27 | \DeclareMathOperator{\EV}{\mathbb E} 28 | \DeclareMathOperator{\Var}{\mathbb V} 29 | 30 | 31 | \begin{document} 32 | \maketitle 33 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 34 | \item Compute the moment generating function of the discrete random variable $X$ with pmf $f(x) = \dfrac{1}{2^x}$, $x = 1, 2, \ldots$. 35 | 36 | \item If $0 < p < 1$ and $q = 1 - p$, find the mgf of the random variable $X$ with pmf $f(x) = p^{x - 1} q$, $x = 1, 2, \ldots$, and hence find $E[X]$ and $V[X]$. 37 | 38 | \item If $E[X^n] = 2^n (n + 1)!$, $n = 0, 1, \ldots$, then find the mgf of $X$. 39 | 40 | \item Let $X$ be a random variable having pdf $f(x) = a e^{-a(x - b)}$, $x \ge b$, where $a > 0$. Using the mgf of $X$, determine its mean and variance. 41 | 42 | \item If $X$ is a random variable with pdf $f(x) = e^{-2|x|}$, find the mgf of $X$, and hence compute $E[X]$ and $V[X]$. 43 | 44 | \item If $X \sim U[a, b]$, compute $E[X^n]$ using $M_X(t)$. Hence show that if $X \sim U[-a, a]$, then $E[X^{2n}] = \dfrac{a^{2n}}{2n + 1}$. 45 | 46 | \item If $M_{X_1}(t) = e^{3t + 2t^2}$, $M_{X_2}(t) = e^{5t + 18t^2}$, $M_{X_3}(t) = e^{4t + 8t^2}$, then find the pdf of $Y = 2X_1 + 3X_2 + 4X_3$, given that $X_1$, $X_2$, and $X_3$ are independent. 47 | 48 | \item If $X \sim N(0, 2)$, then find the moment generating function of $Y = \frac{X^2}{2}$. 49 | 50 | \item Find the mean of $X$, given that its mgf is $M_X(t) = e^{2(e^t - 1)}$. 51 | 52 | \item Find the variance of $X$, given that its mgf is $M_X(t) = \left(\frac 3 4 + \frac{e^t}{4}\right)^{20}$. 53 | 54 | \item Show that if $X$ and $Y$ are independent Poisson variate with means $\lambda$ and $\mu$ respectively, then $X + Y$ is a Poisson variate with mean $\lambda + \mu$. 55 | 56 | \item Show that if $X_i \sim N(\mu_i, \sigma_i^2)$, $i = 1, \ldots, n$ are $n$ independent normal variates, and $a_i$, $i = 1, \ldots, n$ are constants, then $X = \sum_{i=1}^{n} a_i X_i \sim N(\mu, \sigma^2)$ where $\mu = \sum_{i = 1}^{n} a_i \mu_i$ and $\sigma = \sqrt{\sum_{i = 1}^n a_i^2 \sigma_i^2}$. Also determine the distribution of $\overline X = \frac X n$. 57 | 58 | \item Show that if $Z_i \in N(0, 1)$, $i = 1, \ldots, n$ are $n$ independent standard normal variates, then $\sum_{i = 1}^n Z_i^2 \sim \chi^2_n$. 59 | 60 | \end{enumerate} 61 | 62 | \end{document} -------------------------------------------------------------------------------- /Notes/Theory of Computation/Preamble.tex: -------------------------------------------------------------------------------- 1 | % !TeX root = Theory of Computation.tex 2 | 3 | \usepackage{natbib} 4 | 5 | \usepackage{amsmath, amsfonts, amssymb, amsthm, bbm} 6 | \usepackage{tipa} 7 | \usepackage[all, cmtip, 2cell]{xy} 8 | \usepackage{graphicx, subcaption} 9 | \usepackage{physics} 10 | 11 | \usepackage{tikz} 12 | \usetikzlibrary{automata, positioning} 13 | \tikzstyle{accepting} = [double distance = 1.3pt, outer sep = 1pt+\pgflinewidth] 14 | 15 | \usepackage{mathtools} 16 | \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} 17 | \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} 18 | 19 | \usepackage{xspace, cancel} 20 | 21 | \usepackage{enumitem, array} 22 | \setlist{noitemsep} 23 | 24 | \usepackage{scrlayer-scrpage} 25 | \ohead{\color{blue!35!black} \scshape VM} 26 | \cfoot*{\pagemark} 27 | 28 | \setcounter{tocdepth}{3} 29 | \usepackage{tocloft} 30 | \renewcommand{\cftdot}{} 31 | 32 | \usepackage{hyperref} 33 | \definecolor{linkcolor}{RGB}{32, 96, 192} 34 | \hypersetup{colorlinks, linkcolor = linkcolor, urlcolor = linkcolor, linktocpage = true} 35 | \usepackage{bookmark} 36 | \bookmarksetup{color = [RGB]{32, 96, 192}} 37 | 38 | \usepackage[capitalise]{cleveref} 39 | \crefformat{enumi}{#2\textup{#1}#3} 40 | 41 | \usepackage{kbordermatrix} 42 | \usepackage{algorithm, algpseudocode} 43 | 44 | \usepackage{ocgx2} 45 | 46 | \usepackage[intoc]{nomencl} 47 | \makenomenclature 48 | 49 | \usepackage[toc, page]{appendix} 50 | 51 | \usepackage{newpxmath} 52 | \usepackage{charter} 53 | \usepackage[T1]{fontenc} 54 | 55 | \newtheorem{Theorem}{Theorem}[section] 56 | \newtheorem{Lemma}[Theorem]{Lemma} 57 | \newtheorem{Corollary}[Theorem]{Corollary} 58 | \newtheorem{Observation}[Theorem]{Observation} 59 | 60 | \theoremstyle{definition} 61 | \newtheorem{Definition}[Theorem]{Definition} 62 | \newtheorem*{Definition*}{Definition} 63 | \newtheorem{Example}[Theorem]{Example} 64 | \newtheorem*{Example*}{Example} 65 | \newtheorem{Exercise}{Exercise}[section] 66 | 67 | \theoremstyle{remark} 68 | \newtheorem*{Remark*}{Remark} 69 | \newtheorem*{Note*}{Note} 70 | \newtheorem*{Solution*}{Solution} 71 | 72 | \newcommand{\solution}[2]{ 73 | \noindent\textit{\switchocg{sol:#1}{Solution.}} \begin{ocg}{Solution}{sol:#1}{0} 74 | #2 75 | \end{ocg} 76 | } 77 | 78 | \crefname{Section}{Section}{Sections} 79 | \crefname{Exercise}{Exercise}{Exercises} 80 | 81 | \definecolor{alertcolor}{rgb}{0.8, 0.5, 0} 82 | \newcommand{\newterm}[1]{{\color{alertcolor} #1}} 83 | 84 | \definecolor{hintcolor}{rgb}{0, 0, 0.5} 85 | \newcommand{\hint}[2]{ 86 | \switchocg{hint:#1}{\color{hintcolor}\textbf{\small Hint.}} 87 | \begin{ocg}{Hint}{hint:#1}{0} 88 | \small \color{hintcolor} #2 89 | \end{ocg} 90 | } 91 | 92 | \newcommand{\rev}{^\mathrm{R}} 93 | \DeclareMathOperator{\Nb}{N} 94 | \DeclareMathOperator{\lcm}{lcm} 95 | \DeclareMathOperator{\im}{im} 96 | \DeclareMathOperator{\dom}{dom} 97 | \DeclareMathOperator{\cod}{cod} 98 | \newcommand{\ttt}{\texttt} 99 | \newcommand{\tz}{\ttt 0} 100 | \newcommand{\tone}{\ttt 1} 101 | \newcommand{\ta}{\ttt a} 102 | \newcommand{\tb}{\ttt b} 103 | \newcommand{\tc}{\ttt c} 104 | \newcommand{\id}{\mathrm{id}} 105 | \newcommand{\symdiff}{\mathbin{\triangle}} 106 | \newcommand{\nth}{\textsuperscript{th}\xspace} 107 | 108 | \algrenewcommand\algorithmicrequire{\textbf{Input:}} -------------------------------------------------------------------------------- /Notes/Applied Graph Theory/Applied Graph Theory.out.ps: -------------------------------------------------------------------------------- 1 | %! 2 | /pdfmark where{pop} 3 | {/globaldict where{pop globaldict}{userdict}ifelse/pdfmark/cleartomark load put} 4 | ifelse 5 | [ 6 | /Title(\376\377\000L\000i\000s\000t\000\040\000o\000f\000\040\000S\000y\000m\000b\000o\000l\000s) 7 | /C[.12549 .37646 .75294] 8 | /Action/GoTo/Dest(section*.1)cvn 9 | /OUT pdfmark 10 | [ 11 | /Title(\376\377\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n) 12 | /C[.12549 .37646 .75294] 13 | /Action/GoTo/Dest(section.1)cvn 14 | /OUT pdfmark 15 | [ 16 | /Title(\376\377\000C\000a\000r\000t\000e\000s\000i\000a\000n\000\040\000P\000r\000o\000d\000u\000c\000t\000s) 17 | /C[.12549 .37646 .75294] 18 | /Action/GoTo/Dest(section.2)cvn 19 | /OUT pdfmark 20 | [ 21 | /Title(\376\377\000S\000u\000b\000g\000r\000a\000p\000h\000s\000\040\000a\000n\000d\000\040\000C\000o\000m\000p\000l\000e\000m\000e\000n\000t\000s) 22 | /C[.12549 .37646 .75294] 23 | /Action/GoTo/Dest(section.3)cvn 24 | /OUT pdfmark 25 | [ 26 | /Title(\376\377\000W\000a\000l\000k\000s\000,\000\040\000P\000a\000t\000h\000s\000,\000\040\000a\000n\000d\000\040\000C\000y\000c\000l\000e\000s) 27 | /C[.12549 .37646 .75294] 28 | /Action/GoTo/Dest(section.4)cvn 29 | /OUT pdfmark 30 | [ 31 | /Title(\376\377\000D\000i\000s\000t\000a\000n\000c\000e\000s) 32 | /C[.12549 .37646 .75294] 33 | /Action/GoTo/Dest(section.5)cvn 34 | /OUT pdfmark 35 | [ 36 | /Title(\376\377\000C\000o\000n\000n\000e\000c\000t\000e\000d\000n\000e\000s\000s) 37 | /C[.12549 .37646 .75294] 38 | /Action/GoTo/Dest(section.6)cvn 39 | /OUT pdfmark 40 | [ 41 | /Title(\376\377\000G\000r\000a\000p\000h\000\040\000I\000s\000o\000m\000o\000r\000p\000h\000i\000s\000m) 42 | /C[.12549 .37646 .75294] 43 | /Action/GoTo/Dest(section.7)cvn 44 | /OUT pdfmark 45 | [ 46 | /Title(\376\377\000S\000e\000l\000f\000-\000C\000o\000m\000p\000l\000e\000m\000e\000n\000t\000a\000r\000y\000\040\000G\000r\000a\000p\000h\000s) 47 | /C[.12549 .37646 .75294] 48 | /Action/GoTo/Dest(section.8)cvn 49 | /OUT pdfmark 50 | [ 51 | /Title(\376\377\000B\000i\000p\000a\000r\000t\000i\000t\000e\000\040\000G\000r\000a\000p\000h\000s) 52 | /C[.12549 .37646 .75294] 53 | /Action/GoTo/Dest(section.9)cvn 54 | /OUT pdfmark 55 | [ 56 | /Title(\376\377\000T\000r\000e\000e\000s) 57 | /C[.12549 .37646 .75294] 58 | /Action/GoTo/Dest(section.10)cvn 59 | /OUT pdfmark 60 | [ 61 | /Title(\376\377\000B\000l\000o\000c\000k\000s) 62 | /C[.12549 .37646 .75294] 63 | /Action/GoTo/Dest(section.11)cvn 64 | /OUT pdfmark 65 | [ 66 | /Title(\376\377\000L\000i\000n\000e\000\040\000G\000r\000a\000p\000h\000s) 67 | /C[.12549 .37646 .75294] 68 | /Action/GoTo/Dest(section.12)cvn 69 | /OUT pdfmark 70 | [ 71 | /Title(\376\377\000A\000d\000j\000a\000c\000e\000n\000c\000y\000\040\000M\000a\000t\000r\000i\000c\000e\000s) 72 | /C[.12549 .37646 .75294] 73 | /Action/GoTo/Dest(section.13)cvn 74 | /OUT pdfmark 75 | [ 76 | /Title(\376\377\000I\000n\000c\000i\000d\000e\000n\000c\000e\000\040\000M\000a\000t\000r\000i\000c\000e\000s) 77 | /C[.12549 .37646 .75294] 78 | /Action/GoTo/Dest(section.14)cvn 79 | /OUT pdfmark 80 | [ 81 | /Title(\376\377\000D\000i\000j\000k\000s\000t\000r\000a\000'\000s\000\040\000A\000l\000g\000o\000r\000i\000t\000h\000m) 82 | /C[.12549 .37646 .75294] 83 | /Action/GoTo/Dest(section.15)cvn 84 | /OUT pdfmark 85 | [ 86 | /Title(\376\377\000P\000r\000i\000m\000'\000s\000\040\000A\000l\000g\000o\000r\000i\000t\000h\000m) 87 | /C[.12549 .37646 .75294] 88 | /Action/GoTo/Dest(section.16)cvn 89 | /OUT pdfmark 90 | -------------------------------------------------------------------------------- /Notes/MAT 2138 - Discrete Mathematics/Preamble.tex: -------------------------------------------------------------------------------- 1 | % !TeX root = MAT 2138 - Discrete Mathematics.tex 2 | 3 | \usepackage{natbib} 4 | 5 | \usepackage{amsmath, amsfonts, amssymb, amsthm, bbm} 6 | \usepackage{tipa} 7 | \usepackage[all, cmtip, 2cell]{xy} 8 | \usepackage{graphicx, subcaption} 9 | \usepackage{physics} 10 | 11 | \usepackage{tikz} 12 | \usetikzlibrary{automata, positioning} 13 | \tikzstyle{accepting} = [double distance = 1.3pt, outer sep = 1pt+\pgflinewidth] 14 | 15 | \usepackage{mathtools} 16 | \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} 17 | \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} 18 | 19 | \usepackage{xspace, cancel} 20 | 21 | \usepackage{enumitem, array} 22 | \setlist{noitemsep} 23 | 24 | \usepackage{scrlayer-scrpage} 25 | \ohead{\color{blue!35!black} \scshape VM} 26 | \cfoot*{\pagemark} 27 | 28 | \setcounter{tocdepth}{3} 29 | \usepackage{tocloft} 30 | \renewcommand{\cftdot}{} 31 | 32 | \usepackage{hyperref} 33 | \definecolor{linkcolor}{RGB}{32, 96, 192} 34 | \hypersetup{colorlinks, linkcolor = linkcolor, urlcolor = linkcolor, linktocpage = true} 35 | \usepackage{bookmark} 36 | \bookmarksetup{color = [RGB]{32, 96, 192}} 37 | 38 | \usepackage[capitalise]{cleveref} 39 | \crefformat{enumi}{#2\textup{#1}#3} 40 | 41 | \usepackage{kbordermatrix} 42 | \usepackage{algorithm, algpseudocode} 43 | 44 | \usepackage{ocgx2} 45 | 46 | \usepackage[intoc]{nomencl} 47 | \makenomenclature 48 | 49 | \usepackage[toc, page]{appendix} 50 | 51 | \usepackage{newpxmath} 52 | \usepackage{charter} 53 | \usepackage[T1]{fontenc} 54 | 55 | \newtheorem{Theorem}{Theorem}[section] 56 | \newtheorem{Lemma}[Theorem]{Lemma} 57 | \newtheorem{Corollary}[Theorem]{Corollary} 58 | \newtheorem{Observation}[Theorem]{Observation} 59 | 60 | \theoremstyle{definition} 61 | \newtheorem{Definition}[Theorem]{Definition} 62 | \newtheorem*{Definition*}{Definition} 63 | \newtheorem{Example}[Theorem]{Example} 64 | \newtheorem*{Example*}{Example} 65 | \newtheorem{Exercise}{Exercise}[section] 66 | 67 | \theoremstyle{remark} 68 | \newtheorem*{Remark*}{Remark} 69 | \newtheorem*{Note*}{Note} 70 | \newtheorem*{Solution*}{Solution} 71 | 72 | \newcommand{\solution}[2]{ 73 | \noindent\textit{\switchocg{sol:#1}{Solution.}} \begin{ocg}{Solution}{sol:#1}{0} 74 | #2 75 | \end{ocg} 76 | } 77 | 78 | \crefname{Section}{Section}{Sections} 79 | \crefname{Exercise}{Exercise}{Exercises} 80 | 81 | \definecolor{alertcolor}{rgb}{0.8, 0.5, 0} 82 | \newcommand{\newterm}[1]{{\color{alertcolor} #1}} 83 | 84 | \definecolor{hintcolor}{rgb}{0, 0, 0.5} 85 | \newcommand{\hint}[2]{ 86 | \switchocg{hint:#1}{\color{hintcolor}\textbf{\small Hint.}} 87 | \begin{ocg}{Hint}{hint:#1}{0} 88 | \small \color{hintcolor} #2 89 | \end{ocg} 90 | } 91 | 92 | \DeclareMathOperator{\Nb}{N} 93 | \DeclareMathOperator{\lcm}{lcm} 94 | \DeclareMathOperator{\im}{im} 95 | \DeclareMathOperator{\dom}{dom} 96 | \DeclareMathOperator{\cod}{cod} 97 | \newcommand{\ttt}{\texttt} 98 | \newcommand{\tz}{\ttt 0} 99 | \newcommand{\tone}{\ttt 1} 100 | \newcommand{\ta}{\ttt a} 101 | \newcommand{\tb}{\ttt b} 102 | \newcommand{\tc}{\ttt c} 103 | \DeclareMathOperator{\ecc}{ecc} 104 | \DeclareMathOperator{\rad}{rad} 105 | \DeclareMathOperator{\diam}{diam} 106 | \newcommand{\id}{\mathrm{id}} 107 | \newcommand{\symdiff}{\mathbin{\triangle}} 108 | \newcommand{\nth}{\textsuperscript{th}\xspace} 109 | \DeclareMathOperator{\bestDTo}{bestDTo} 110 | \DeclareMathOperator{\tree}{tree} 111 | 112 | \algrenewcommand\algorithmicrequire{\textbf{Input:}} -------------------------------------------------------------------------------- /Problem Sets/MAT 2155 - Combinatorics/Combinatorics - Set 2.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = xelatex 2 | 3 | \documentclass[svgnames]{amsart} 4 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[T1]{fontenc} 12 | \usepackage[sfdefault, lining, scale=0.9]{FiraSans} 13 | \usepackage{kmath} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | \linespread{1.2} 24 | 25 | \title{MAT 2155: Problem Set 2} 26 | \date{} 27 | 28 | \begin{document} 29 | \maketitle 30 | \begin{enumerate}[leftmargin=*] 31 | \item Number of rearrangements of the word \texttt{SESQUIPEDALIAN}. 32 | 33 | \item Number of anagrams of the word \texttt{INDEPENDENCE} which 34 | \begin{enumerate}[label=(\roman*)] 35 | \item start with the letter \texttt{C}. 36 | \item all the vowels occur together. 37 | \item no two vowels occur together. 38 | \item begin in \texttt{I} and end in \texttt{E}. 39 | \end{enumerate} 40 | 41 | \item Number of positive integers less than $1000$ that have distinct digits. 42 | 43 | \item Number of ways of choosing $4$ cards from a pack of $52$ playing cards 44 | \begin{enumerate}[label=(\roman*)] 45 | \item with no restrictions. 46 | \item with all four cards from the same suit. 47 | \item with all four cards from different suits. 48 | \item such that all four cards are face cards. 49 | \item such that $2$ are red and $2$ are black. 50 | \item such that all four cards are of the same colour. 51 | \end{enumerate} 52 | 53 | \item There are $9$ different books on a bookshelf, $4$ red and $5$ green. In how many different ways can the books be arranged if 54 | \begin{enumerate}[label=(\roman*)] 55 | \item there are no restrictions? 56 | \item the red books must be together, and the green books must be together? 57 | \item the red books must be together? 58 | \item no two books of the same colour must be together? 59 | \end{enumerate} 60 | 61 | \item A shop sells six different flavours of ice cream. In how many ways can a customer choose $4$ ice cream cones 62 | \begin{enumerate}[label=(\roman*)] 63 | \item all of different flavours? 64 | \item not necessarily of different flavours? 65 | \item of $2$ or $3$ different flavours? 66 | \item exactly $3$ different flavours? 67 | \end{enumerate} 68 | 69 | \item Number of ways of buying a total of $7$ fruits from a shop selling apples, oranges, and strawberries, if you must buy at least one fruit of each type. 70 | 71 | \item Number of ways of selecting $10$ marbles from a pile of red, blue, and green marbles if there must be 72 | \begin{enumerate}[label=(\roman*)] 73 | \item at least $5$ red marbles. 74 | \item at most $5$ red marbles. 75 | \end{enumerate} 76 | 77 | \item Number of integer solutions of the equation $x_1 + x_2 + x_3 + x_4 = 18$ with 78 | \begin{enumerate}[label=(\roman*)] 79 | \item $x_i \ge 0$, $i = 1, 2, 3, 4$. 80 | \item $x_i \ge 1$, $i = 1, 2, 3, 4$. 81 | \item $x_i \ge i$, $i = 1, 2, 3, 4$. 82 | \end{enumerate} 83 | 84 | \item Number of binary sequences of length $10$ consisting of a run of $1$s followed by a run of $0$s followed by a run of $1$s followed by another run of $0$s. 85 | 86 | \item Number of ways to distribute $15$ identical objects into $4$ different boxes such that the number of objects in the fourth box is a multiple of $3$. 87 | \end{enumerate} 88 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/MAT 2155 - Combinatorics/Combinatorics - Set 1.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = xelatex 2 | 3 | \documentclass[svgnames]{amsart} 4 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[T1]{fontenc} 12 | \usepackage[sfdefault, lining, scale=0.9]{FiraSans} 13 | \usepackage{kmath} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | \linespread{1.2} 24 | 25 | \title{MAT 2155: Problem Set 1} 26 | \date{} 27 | 28 | \begin{document} 29 | \maketitle 30 | \begin{enumerate}[leftmargin=*] 31 | \item Number of ways of selecting one left and one right shoe from six pairs of shoes, without obtaining a pair. 32 | 33 | \item Number of ways of picking two books, not both in the same language, from a collection of five different English books, six different German books, and seven different French books. 34 | 35 | \item Number of positive integers less than $1$ million formed using 36 | \begin{enumerate}[label=(\roman*)] 37 | \item $7$s, $8$s, and $9$s only. 38 | \item $0$s, $8$s, and $9$s only. 39 | \end{enumerate} 40 | 41 | \item Sum of all $4$-digit numbers that can be obtained using the digits 42 | \begin{enumerate}[label=(\roman*)] 43 | \item $1$, $2$, $3$, $4$ once each. 44 | \item $0$, $1$, $2$, $3$ once each. 45 | \item $1$, $2$, $3$, $4$. 46 | \item $0$, $1$, $2$, $3$. 47 | \end{enumerate} 48 | 49 | \item Number of $3$-letter sequences using the letters \emph{a, b, c, d, e, f} 50 | \begin{enumerate}[label=(\roman*)] 51 | \item with repetition allowed. 52 | \item without repetition. 53 | \item without repetition, and containing the letter \emph{e}. 54 | \item with repetition, and containing \emph{e}. 55 | \end{enumerate} 56 | 57 | \item Five people A, B, C, D, E intend to speak at a meeting. In how many ways can they do so 58 | \begin{enumerate}[label=(\roman*)] 59 | \item without B speaking before A? 60 | \item if A must speak immediately before B? 61 | \end{enumerate} 62 | 63 | \item Number of ways of selecting three integers from $3n$ consecutive integers so that their sum is a multiple of $3$. 64 | 65 | \item Number of times the digit $5$ is written when listing all numbers from $1$ to $100,000$. 66 | 67 | \item Number of times $25$ is written when listing all numbers from $1$ to $100,000$. 68 | 69 | \item Number of four-digit numbers divisible be $4$ that can be formed using the digits 70 | \begin{enumerate}[label=(\roman*)] 71 | \item $1$, $2$, $3$, $4$, $5$ with possible repetition. 72 | \item $1$, $2$, $3$, $4$, $5$ without repetition. 73 | \item $0$, $1$, $2$, $3$, $4$ with possible repetition. 74 | \item $0$, $1$, $2$, $3$, $4$ without repetition. 75 | \end{enumerate} 76 | 77 | \item Number of ways of wearing five rings on four fingers (not including the thumb) of your right hand. 78 | 79 | \item Six distinct symbols are transmitted through a communication channel. A total of twelve blanks are to be inserted between the symbols with at least two blanks between every pair of symbols. In how many ways can we arrange the symbols and blanks? 80 | 81 | \item Number of ways of placing two black queens on an $8 \times 8$ chessboard so that they are not attacking each other. 82 | 83 | \item How many points of intersection are formed by $n$ lines drawn in a plane if no two are parallel and no three concurrent? Into how many regions is the plane divided? 84 | 85 | \item If a convex decagon is such that no three of its diagonals meet at the same point inside it, then how many segments are the diagonals divided into by their intersections? 86 | \end{enumerate} 87 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 7.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx, tikz} 6 | \usepackage{mathtools, physics} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[default,bold]{sourceserifpro} 12 | \usepackage[T1]{fontenc} 13 | \usepackage[]{fouriernc} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\scriptsize\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | 24 | \title[]{Probability and Statistics -- Problem Set 7} 25 | 26 | \DeclareMathOperator{\Prob}{P} 27 | \DeclareMathOperator{\EV}{\mathbb E} 28 | \DeclareMathOperator{\Var}{\mathbb V} 29 | 30 | 31 | \begin{document} 32 | \maketitle 33 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 34 | \item If $X$ is a discrete random variable with pmf $f(x)$, compute the pmf $g(y)$ of $Y = X^2 - 1$ in each of the following cases: 35 | \begin{enumerate}[itemsep=5pt] 36 | \item $f(x) = \dfrac 1 3$, $x = 1, 2, 3$. 37 | \item $f(x) = \dfrac 1 3$, $x = -1, 0, 1$. 38 | \item $f(x) = \dfrac{|x|}{2}$, $x = -1, 0, 1$. 39 | \end{enumerate} 40 | 41 | \item If $X \sim U[-1, 1]$, determine the pdfs of $Y = \sin \dfrac{\pi x}{2}$ and $Z = \cos \dfrac{\pi x}{2}$. 42 | 43 | \item Find the pdf of $Y = - \log X^4$ in each of the following cases: 44 | \begin{enumerate} 45 | \item $X \sim U[0, 1]$. 46 | \item $X$ has pdf $f(x) = 4x^3$, $0 < x < 1$. 47 | \end{enumerate} 48 | 49 | \item Show that if $X$ follows the Cauchy distribution with pdf $f(x) = \dfrac{1}{\pi (1 + x^2)}$, then so does $Y = \dfrac 1 X$. 50 | 51 | \item Compute the pdf of $Y = \tan X$, if $X \sim U\bqty{-\frac \pi 2, \frac \pi 2}$. 52 | 53 | \item If $X \sim U[-1, 1]$, find the pdf of 54 | \begin{enumerate} 55 | \item $Y = X^3$. 56 | \item $Z = X^4$. 57 | \item $W = X^n$, where $n$ is any positive integer. 58 | \end{enumerate} 59 | 60 | \item Compute the pdf of $Y = X^2$ if $X \sim U[-1, 2]$. 61 | 62 | \item If $X$ is a random variable with pdf $f(x) = 2x$, $0 < x < 1$, compute the pdf of $Y = e^{-X}$. 63 | 64 | \item Let $X$ be a random variable with pdf $f(x) = \dfrac 1 {2x^2}$, $|x| > 1$. Then show that $Y = \log X^2$ has an exponential distribution. What is the mean of $Y$? 65 | 66 | \item $(X, Y)$ is a two dimensional random variable having joint pdf $f(x, y) = 3xe^{-(x+3y)}$, $x,y > 0$. If $Z = X$ and $W = 2X + Y$, determine the distribution of $(Z, W)$. 67 | 68 | \item Let $(X, Y)$ be uniformly distributed in the unit square $0 \le x, y \le 1$. Find the pdf of $Z = X + Y$. 69 | 70 | \item If $X \sim \mathcal E(2)$ and $Y \sim \mathcal E(1)$ are independent, find the pdf of: 71 | \begin{enumerate} 72 | \item $Z = X + Y$. 73 | \item $Z = X/Y$. 74 | \end{enumerate} 75 | 76 | \item If $(X, Y)$ has the joint pdf $f(x, y) = 10xy^2$, $0 < x < y < 1$, determine the pdf of $Z = X/Y$. 77 | 78 | \item Find the pdf of $Z = X/Y$, if $(X, Y)$ has joint pdf $f(x, y) = 8xy$, $0 < x < y < 1$. 79 | 80 | \item $(X, Y)$ is uniformly distributed over the unit disc $x^2 + y^2 \le 1$. Find the pdf of $R = \sqrt{X^2 + Y^2}$. 81 | 82 | \item Let $X$ be a random variable with pdf $f(x) = \dfrac 5 {x^2}$, $x > 5$. If $X_1$ and $X_2$ are two independent random variables following this distribution, find the pdf of $Y = X_1/X_2$. 83 | 84 | \item Let $W \sim N(0, 1)$ and $V \sim \chi^2_n$. Compute the distribution of $T = \dfrac{W}{\sqrt{V/n}}$. 85 | 86 | \item If $(X_1, X_2)$ has the joint pdf $f(x_1, x_2) = 2e^{-(x_1 + x_2)}$, $x_1 > x_2 > 0$, find the joint pdf of $Y_1 = X_1 - X_2$, $Y_2 = 2X_2$. 87 | 88 | \item If $(X, Y)$ is a two dimensional random variable with joint pdf $f(x, y) = 24 xy$, $x > 0$, $y > 0$, $x + y < 1$, find the pdf of $Z = XY$. 89 | 90 | \end{enumerate} 91 | 92 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 4.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx, tikz} 6 | \usepackage{mathtools, physics} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[]{fouriernc} 12 | \usepackage[default,bold]{sourceserifpro} 13 | \usepackage[T1]{fontenc} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\scriptsize\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | 24 | \title[]{Probability and Statistics -- Problem Set 4} 25 | 26 | \DeclareMathOperator{\Prob}{P} 27 | \DeclareMathOperator{\EV}{\mathbb E} 28 | \DeclareMathOperator{\Var}{\mathbb V} 29 | 30 | 31 | \begin{document} 32 | \maketitle 33 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 34 | \item $(X, Y)$ is a two dimensional random variable with joint pdf $f(x, y) = x + y$, $0 < x < 1$, $0 < y < 1$. Compute the following. 35 | \begin{enumerate} 36 | \item $\Prob[2X + 2Y < 1]$. 37 | \item $\Prob[2X + 2Y < 3]$. 38 | \item $\Prob[X \ge \frac 1 2]$. 39 | \item $\Prob[X \ge \frac 1 2, Y \ge \frac 1 2]$. 40 | \item $\Prob[Y \ge \frac 1 2 \mid X \ge \frac 1 2]$. 41 | \item $\Prob[X + Y > 1 \mid X \ge \frac 1 2]$. 42 | \end{enumerate} 43 | 44 | \item Let $f(x, y) = x^2 + \dfrac{xy}{3}$, $0 \le x \le 1$, $0 \le y \le 2$. Verify that $f(x, y)$ is a joint pdf, and find $P[X + Y > 1]$. 45 | 46 | \item Let $f(x, y) = kx(x - y)$, $|y| < x \le 2$. Find the value $k$ such that $f(x, y)$ is a joint pdf, and compute the marginal pdfs of $X$ and $Y$. 47 | 48 | \item From a box containing $r$ red, $g$ green, and $b$ blue marbles, $n$ marbles are drawn successively \textbf{with replacement}. Let $X$ and $Y$ respectively be the number of blue marbles and number of green marbles obtained in these $n$ draws. 49 | \begin{enumerate} 50 | \item Determine the joint pmf of $(X, Y)$ and compute the marginal pmfs. 51 | \item Are $X$ and $Y$ independent? 52 | \item For the particular case $r = 3$, $g = b = 2$, tabulate the joint pmf values and compute the coefficient of correlation between $X$ and $Y$. 53 | \end{enumerate} 54 | 55 | \item The joint pdf of $(X, Y)$ is $f(x, y) = 2 e^{-x - 2y}$, $x, y > 0$. Compute 56 | \begin{enumerate} 57 | \item $\Prob[X > 1, Y < 1]$. 58 | \item $\Prob[X < Y]$. 59 | \item $\Prob[X < a]$. 60 | \end{enumerate} 61 | 62 | \item Compute the joint pdf of the random variable $(X, Y)$ that is uniformly distributed over the region bounded by the curves $y = x^2$ and $y = x$. Also find the marginal pdfs of $X$ and $Y$. 63 | 64 | \item If $f(x, y) = 2(x + y - 2xy)$, $0 \le x, y \le 1$, show that $Y \sim U[0, 1]$. 65 | 66 | \item Compute the marginal pdfs of $X$ and $Y$ and show that they are independent, given their joint pdf $(X, Y)$. 67 | \begin{enumerate} 68 | \item $f(x, y) = 3 - 6x - y + 2xy$, $0 < x < \frac 1 2$, $0 < y < 2$. 69 | \item $f(x, y) = e^{-2x - \frac y 2}$, $x, y > 0$. 70 | \end{enumerate} 71 | 72 | \item For each random variable $(X, Y)$ with joint pdf $f(x, y)$ as given below, compute the coefficient of correlation $\rho$. 73 | \begin{enumerate} 74 | \item $f(x, y) = x + y$, $0 < x, y < 1$. 75 | \item $f(x, y) = 2x + 2y$, $0 < y < x < 1$. 76 | \item $f(x, y) = 8xy$, $0 < x < y < 1$. 77 | \item $f(x, y) = 2x - xy$, $0 < x < 1$, $0 < y < 2$. 78 | \item $f(x, y) = \dfrac 1 \pi$, $x^2 + y^2 \le 1$ [Uniform distribution in the unit disc centred at the origin]. 79 | \item $f(x, y) = \dfrac 2 \pi$, $y \ge 0$, $x^2 + y^2 \le 1$. 80 | \item $f(x, y) = \dfrac 4 \pi$, $x,y \ge 0$, $x^2 + y^2 \le 1$. 81 | \item $f(x, y) = \dfrac 1 4$, $(x, y) \in R$, where $R$ is the square with vertices $(1, 1)$, $(-1, 1)$, $(-1, -1)$, $(1, -1)$. 82 | \item $f(x, y) = \dfrac 1 2$, $(x, y) \in R$, where $R$ is the square with vertices $(1, 0)$, $(0, 1)$, $(-1, 0)$, $(0, -1)$. 83 | \end{enumerate} 84 | Are $X$ and $Y$ independent in all the cases where $\rho = 0$? 85 | \end{enumerate} 86 | \end{document} -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | ## Core latex/pdflatex auxiliary files: 2 | *.aux 3 | *.lof 4 | *.log 5 | *.lot 6 | *.fls 7 | *.out 8 | *.toc 9 | *.fmt 10 | *.fot 11 | *.cb 12 | *.cb2 13 | .*.lb 14 | 15 | ## Intermediate documents: 16 | *.dvi 17 | *.xdv 18 | *-converted-to.* 19 | # these rules might exclude image files for figures etc. 20 | # *.ps 21 | # *.eps 22 | # *.pdf 23 | 24 | ## Generated if empty string is given at "Please type another file name for output:" 25 | .pdf 26 | 27 | ## Bibliography auxiliary files (bibtex/biblatex/biber): 28 | *.bbl 29 | *.bcf 30 | *.blg 31 | *-blx.aux 32 | *-blx.bib 33 | *.run.xml 34 | 35 | ## Build tool auxiliary files: 36 | *.fdb_latexmk 37 | *.synctex 38 | *.synctex(busy) 39 | *.synctex.gz 40 | *.synctex.gz(busy) 41 | *.pdfsync 42 | 43 | ## Build tool directories for auxiliary files 44 | # latexrun 45 | latex.out/ 46 | 47 | ## Auxiliary and intermediate files from other packages: 48 | # algorithms 49 | *.alg 50 | *.loa 51 | 52 | # achemso 53 | acs-*.bib 54 | 55 | # amsthm 56 | *.thm 57 | 58 | # beamer 59 | *.nav 60 | *.pre 61 | *.snm 62 | *.vrb 63 | 64 | # changes 65 | *.soc 66 | 67 | # comment 68 | *.cut 69 | 70 | # cprotect 71 | *.cpt 72 | 73 | # elsarticle (documentclass of Elsevier journals) 74 | *.spl 75 | 76 | # endnotes 77 | *.ent 78 | 79 | # fixme 80 | *.lox 81 | 82 | # feynmf/feynmp 83 | *.mf 84 | *.mp 85 | *.t[1-9] 86 | *.t[1-9][0-9] 87 | *.tfm 88 | 89 | #(r)(e)ledmac/(r)(e)ledpar 90 | *.end 91 | *.?end 92 | *.[1-9] 93 | *.[1-9][0-9] 94 | *.[1-9][0-9][0-9] 95 | *.[1-9]R 96 | *.[1-9][0-9]R 97 | *.[1-9][0-9][0-9]R 98 | *.eledsec[1-9] 99 | *.eledsec[1-9]R 100 | *.eledsec[1-9][0-9] 101 | *.eledsec[1-9][0-9]R 102 | *.eledsec[1-9][0-9][0-9] 103 | *.eledsec[1-9][0-9][0-9]R 104 | 105 | # glossaries 106 | *.acn 107 | *.acr 108 | *.glg 109 | *.glo 110 | *.gls 111 | *.glsdefs 112 | *.lzo 113 | *.lzs 114 | 115 | # uncomment this for glossaries-extra (will ignore makeindex's style files!) 116 | # *.ist 117 | 118 | # gnuplottex 119 | *-gnuplottex-* 120 | 121 | # gregoriotex 122 | *.gaux 123 | *.gtex 124 | 125 | # htlatex 126 | *.4ct 127 | *.4tc 128 | *.idv 129 | *.lg 130 | *.trc 131 | *.xref 132 | 133 | # hyperref 134 | *.brf 135 | 136 | # knitr 137 | *-concordance.tex 138 | # TODO Comment the next line if you want to keep your tikz graphics files 139 | *.tikz 140 | *-tikzDictionary 141 | 142 | # listings 143 | *.lol 144 | 145 | # luatexja-ruby 146 | *.ltjruby 147 | 148 | # makeidx 149 | *.idx 150 | *.ilg 151 | *.ind 152 | 153 | # minitoc 154 | *.maf 155 | *.mlf 156 | *.mlt 157 | *.mtc[0-9]* 158 | *.slf[0-9]* 159 | *.slt[0-9]* 160 | *.stc[0-9]* 161 | 162 | # minted 163 | _minted* 164 | *.pyg 165 | 166 | # morewrites 167 | *.mw 168 | 169 | # nomencl 170 | *.nlg 171 | *.nlo 172 | *.nls 173 | 174 | # pax 175 | *.pax 176 | 177 | # pdfpcnotes 178 | *.pdfpc 179 | 180 | # sagetex 181 | *.sagetex.sage 182 | *.sagetex.py 183 | *.sagetex.scmd 184 | 185 | # scrwfile 186 | *.wrt 187 | 188 | # sympy 189 | *.sout 190 | *.sympy 191 | sympy-plots-for-*.tex/ 192 | 193 | # pdfcomment 194 | *.upa 195 | *.upb 196 | 197 | # pythontex 198 | *.pytxcode 199 | pythontex-files-*/ 200 | 201 | # tcolorbox 202 | *.listing 203 | 204 | # thmtools 205 | *.loe 206 | 207 | # TikZ & PGF 208 | *.dpth 209 | *.md5 210 | *.auxlock 211 | 212 | # todonotes 213 | *.tdo 214 | 215 | # vhistory 216 | *.hst 217 | *.ver 218 | 219 | # easy-todo 220 | *.lod 221 | 222 | # xcolor 223 | *.xcp 224 | 225 | # xmpincl 226 | *.xmpi 227 | 228 | # xindy 229 | *.xdy 230 | 231 | # xypic precompiled matrices and outlines 232 | *.xyc 233 | *.xyd 234 | 235 | # endfloat 236 | *.ttt 237 | *.fff 238 | 239 | # Latexian 240 | TSWLatexianTemp* 241 | 242 | ## Editors: 243 | # WinEdt 244 | *.bak 245 | *.sav 246 | 247 | # Texpad 248 | .texpadtmp 249 | 250 | # LyX 251 | *.lyx~ 252 | 253 | # Kile 254 | *.backup 255 | 256 | # gummi 257 | .*.swp 258 | 259 | # KBibTeX 260 | *~[0-9]* 261 | 262 | # auto folder when using emacs and auctex 263 | ./auto/* 264 | *.el 265 | 266 | # expex forward references with \gathertags 267 | *-tags.tex 268 | 269 | # standalone packages 270 | *.sta 271 | 272 | # Makeindex log files 273 | *.lpz 274 | -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 9.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx, tikz} 6 | \usepackage{mathtools, physics} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[default,bold]{sourceserifpro} 12 | \usepackage[T1]{fontenc} 13 | \usepackage[]{fouriernc} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\scriptsize\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | 24 | \title[]{Probability and Statistics -- Problem Set 9} 25 | 26 | \DeclareMathOperator{\Prob}{P} 27 | \DeclareMathOperator{\EV}{\mathbb E} 28 | \DeclareMathOperator{\Var}{\mathbb V} 29 | 30 | 31 | \begin{document} 32 | \maketitle 33 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 34 | \item If $\overline X$ is the mean of a random sample of size $5$ from a normal distribution with $\mu = 0$ and $\sigma^2 = 125$, determine $c$ such that $\Prob\bqty{\overline X < c} = 0.9$. 35 | 36 | \item If $X \sim N(\mu, 100)$, find $n$ such that $\Prob\bqty{|\overline X - \mu| < 5} = 0.954$. 37 | 38 | \item Let $\overline X$ be the mean of a sample of size $75$ from a distribution that has pdf $f(x) = 1$, $0 < x < 1$. Find an approximate probability $\Prob\bqty{0.45 < \overline X < 0.55}$. 39 | 40 | \item Compute $\Prob\bqty{2.3 < S^2 < 22.2}$, where $S^2$ is the variance of a sample of size $6$ from a normal distribution with variance $12$. 41 | 42 | \item If $X \sim N(0, 16))$ and $Y \sim N(1, 9)$ respectively, and $\overline X$ and $\overline Y$ are the means of samples of size $25$ each from these two distributions respectively, find $\Prob\bqty{\overline X > \overline Y}$. 43 | 44 | \item If $X$ has a distribution defined by the pdf $f(x) = 3x^2$, $0 < x < 1$, compute an approximate probability that the mean of a sample of size $15$ taken from this population lies between $\frac 3 5$ and $\frac 4 5$. 45 | 46 | \item If $Y \sim B(n, 0.55)$, find an approximation for the smallest value of $n$ such that $\Prob\bqty{\frac Y n > \frac 1 2} \ge 0.95$. 47 | 48 | \item Let $Y$ be the sum of the outcomes obtained when a fair, six-sided die is rolled $12$ times. Using Central Limit Theorem, compute $\Prob\bqty{36 \le Y \le 48}$ approximately. 49 | 50 | \item Consider a random sample of size $72$ from a distribution having pdf $f(x) = \frac 1 {x^2}$, $x > 1$. What is the probability that more than $50$ of the observations in the sample are less than $3$? 51 | 52 | \item Forty-eight measurements are recorded to several decimal places, and each of these is rounded off to the nearest integer. The sum of the original $48$ measurements is approximated by the sum of these integers. Assuming that the round-off errors are independent and each is uniformly distribution in $(-0.5, 0.5)$, compute an approximate probability that the sum of the integers is within two units of the true sum. 53 | 54 | \item If $X \sim N(10, 9)$ and $Y \sim N(3, 4)$, find $\Prob\bqty{\overline X > 2 \overline Y}$ where both samples have size $4$. 55 | 56 | \item Compute $\Prob\bqty{0 < \overline X < 6, 55.2 < S^2 < 145.6}$, if $X \sim N(3, 100)$ and $n = 25$. 57 | 58 | \item Approximate the probability that the sum of $16$ independent random variables, each uniformly distributed in $[0, 1]$, exceeds $10$. 59 | 60 | \item Compute an approximate probability that $\overline X$ is between $7$ and $9$, if $X \sim \Gamma\pqty{2, \frac 1 4}$ and $n = 128$. 61 | 62 | \item A fair, six-sided die is rolled until the total of all outcomes exceeds $400$. Approximate the probability that this will require more than $140$ rolls. 63 | 64 | \item The lifetime, in hours, of a type of electric bulb has expected value $500$ and standard deviation $80$. Approximate the probability that the mean of the lifetimes of a sample of $n$ such bulbs is greater than $525$, when 65 | \begin{enumerate}[label=(\roman*)] 66 | \item $n = 4$ 67 | \item $n = 16$ 68 | \item $n = 36$ 69 | \item $n = 64$. 70 | \end{enumerate} 71 | 72 | \end{enumerate} 73 | 74 | \end{document} -------------------------------------------------------------------------------- /Obsidian/.obsidian/workspace.json: -------------------------------------------------------------------------------- 1 | { 2 | "main": { 3 | "id": "44aa6c1216c2db91", 4 | "type": "split", 5 | "children": [ 6 | { 7 | "id": "b4841a17c3c69545", 8 | "type": "tabs", 9 | "children": [ 10 | { 11 | "id": "91cff33a845a9fde", 12 | "type": "leaf", 13 | "state": { 14 | "type": "markdown", 15 | "state": { 16 | "file": "Theory of Computation.md", 17 | "mode": "source", 18 | "source": false 19 | } 20 | } 21 | } 22 | ] 23 | } 24 | ], 25 | "direction": "vertical" 26 | }, 27 | "left": { 28 | "id": "6aecb2ee3e5b279b", 29 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amsfonts, amssymb, amsthm} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[T1]{fontenc} 12 | \usepackage[sfdefault, lining, scale=0.9]{FiraSans} 13 | \usepackage{kmath} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | \linespread{1.2} 24 | 25 | \title{MAT 2155: Problem Set 4} 26 | \date{} 27 | 28 | \begin{document} 29 | \maketitle 30 | \begin{enumerate}[leftmargin=*] 31 | \item The coefficient of $x^6$ in the expansion of 32 | \begin{enumerate}[label=(\roman*)] 33 | \item $(1 + x)^8$. 34 | \item $(1 - x)^{-8}$. 35 | \item $(1 + x^2)^4$. 36 | \item $(1 - x^2)^{-4}$. 37 | \item $(1 + x + x^2)^4$. 38 | \item $(1 + x^2)(1 + x^2 + x^4 + \cdots)$. 39 | \end{enumerate} 40 | 41 | \item Using generating functions, find the number of ways of selecting $6$ objects from 42 | \begin{enumerate}[label=(\roman*)] 43 | \item $8$ distinct objects. 44 | \item $8$ types of object, with any number of objects of each type. 45 | \item $4$ types of objects, such that zero or two objects are chosen from each type. 46 | \item $4$ types of objects, such that an even number of objects is chosen from each type. 47 | \item $4$ types of objects, such that not more than two objects are chosen from each type. 48 | \item $2$ types of objects, such that zero or two objects of the first type are chosen, and an even number of objects is chosen from the second type. 49 | \end{enumerate} 50 | 51 | \item Using generating functions, find the number of integer solutions of the equation 52 | \begin{enumerate}[label=(\roman*)] 53 | \item $x_1 + \cdots + x_8 = 6$, $x_i \in \{0, 1\}$, $i = 1, \ldots, 8$. 54 | \item $x_1 + \cdots + x_8 = 6$, $x_i \ge 0$, $i = 1, \ldots, 8$. 55 | \item $x_1 + \cdots + x_4 = 6$, $x_i \in \{0, 2\}$, $i = 1, \ldots, 4$. 56 | \item $x_1 + \cdots + x_4 = 6$, $x_i \ge 0$, $x_i$ is even, $i = 1, \ldots, 4$. 57 | \item $x_1 + \cdots + x_4 = 6$, $0 \le x_i \le 2$, $i = 1, \ldots, 4$. 58 | \item $x + y = 6$, $x \in \{0, 2\}$, $y \ge 0$, $y$ is even. 59 | \end{enumerate} 60 | 61 | \item Number of ways of distributing $30$ identical objects into $3$ distinct boxes such that no box is empty. 62 | 63 | \item Number of ways of distributing $30$ identical marbles into $6$ boxes with at most $10$ marbles in the first box. 64 | 65 | \item Number of ways of selecting $12$ flowers for a bouquet from roses, lilacs, tulips, and lilies, with between $2$ and $5$ of each kind. 66 | 67 | \item Number of ways to select $10$ marbles from a large pile of red, white, and blue marbles if 68 | \begin{enumerate}[label=(\roman*)] 69 | \item the selection has at least $2$ marbles of each colour. 70 | \item the selection has at most $2$ red marbles. 71 | \item the selection has an even number of blue marbles. 72 | \end{enumerate} 73 | 74 | \item Number of ways to place an order of $12$ chocolate sundaes if there are $5$ types of sundaes, and at most $4$ sundaes of one type are allowed. 75 | 76 | \item Number of ways to get a sum of $25$ when $10$ distinct dice are rolled. 77 | 78 | \item Number of ways to select $300$ chocolate candies from $7$ types of candy if each type comes in boxes of $20$, and at least $1$ but not more than $5$ boxes of each type are chosen. 79 | 80 | \item Number of ways of distributing $30$ distinct objects into $3$ boxes such that no box is empty. 81 | 82 | \item Using generating functions, find the number of $r$-permutations of objects chosen from unlimited supplies of $n$ types of objects. 83 | 84 | \item Number of $r$-digit quaternary sequences (with digits $0$, $1$, $2$, $3$) having an even number of $0$s and an odd number of $1$s. 85 | 86 | \item Write the exponential generating function for the number of arrangements of $k$ objects chosen from $n$ types with at most $4$ objects of each type. 87 | 88 | \item Number of $n$-digit ternary sequences with 89 | \begin{enumerate}[label=(\roman*)] 90 | \item an even number of $0$s. 91 | \item an even number of $0$s and an even number of $1$s. 92 | \item $0$ and $1$ occurring a positive even number of times. 93 | \item at least one $0$ and at least one $1$. 94 | \item the total number of $0$s and $1$s being even. 95 | \item no digit occurring exactly twice. 96 | \end{enumerate} 97 | \end{enumerate} 98 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/MAC 1103 - Computational Mathematics/Generating Functions.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = xelatex 2 | 3 | \documentclass[svgnames]{amsart} 4 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[T1]{fontenc} 12 | \usepackage[sfdefault, lining, scale=0.9]{FiraSans} 13 | \usepackage{kmath} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | \linespread{1.2} 24 | 25 | \title{MAC 1103: Generating Functions} 26 | \date{} 27 | 28 | \begin{document} 29 | \maketitle 30 | \begin{enumerate}[leftmargin=*] 31 | \item The coefficient of $x^6$ in the expansion of 32 | \begin{enumerate}[label=(\roman*)] 33 | \item $(1 + x)^8$. 34 | \item $(1 - x)^{-8}$. 35 | \item $(1 + x^2)^4$. 36 | \item $(1 - x^2)^{-4}$. 37 | \item $(1 + x + x^2)^4$. 38 | \item $(1 + x^2)(1 + x + x^2 + \cdots)$. 39 | \end{enumerate} 40 | 41 | \item Using generating functions, find the number of ways of selecting $6$ objects from 42 | \begin{enumerate}[label=(\roman*)] 43 | \item $8$ distinct objects. 44 | \item $8$ types of object, with any number of objects of each type. 45 | \item $4$ types of objects, such that zero or two objects are chosen from each type. 46 | \item $4$ types of objects, such that an even number of objects is chosen from each type. 47 | \item $4$ types of objects, such that not more than two objects are chosen from each type. 48 | \item $2$ types of objects, such that zero or two objects of the first type are chosen, and an even number of objects is chosen from the second type. 49 | \end{enumerate} 50 | 51 | \item Using generating functions, find the number of integer solutions of the equation 52 | \begin{enumerate}[label=(\roman*)] 53 | \item $x_1 + \cdots + x_8 = 6$, $x_i \in \{0, 1\}$, $i = 1, \ldots, 8$. 54 | \item $x_1 + \cdots + x_8 = 6$, $x_i \ge 0$, $i = 1, \ldots, 8$. 55 | \item $x_1 + \cdots + x_4 = 6$, $x_i \in \{0, 2\}$, $i = 1, \ldots, 4$. 56 | \item $x_1 + \cdots + x_4 = 6$, $x_i \ge 0$, $x_i$ is even, $i = 1, \ldots, 4$. 57 | \item $x_1 + \cdots + x_4 = 6$, $0 \le x_i \le 2$, $i = 1, \ldots, 4$. 58 | \item $x + y = 6$, $x \in \{0, 2\}$, $y \ge 0$, $y$ is even. 59 | \end{enumerate} 60 | 61 | \item Number of ways of distributing $30$ identical objects into $3$ distinct boxes such that no box is empty. 62 | 63 | \item Number of ways of distributing $30$ identical marbles into $6$ boxes with at most $10$ marbles in the first box. 64 | 65 | \item Number of ways of selecting $12$ flowers for a bouquet from roses, lilacs, tulips, and lilies, with between $2$ and $5$ of each kind. 66 | 67 | \item Number of ways to select $10$ marbles from a large pile of red, white, and blue marbles if 68 | \begin{enumerate}[label=(\roman*)] 69 | \item the selection as at least $2$ marbles of each colour. 70 | \item the selection as at most $2$ red marbles. 71 | \item the selection has an even number of blue marbles. 72 | \end{enumerate} 73 | 74 | \item Number of ways to place an order of $12$ chocolate sundaes if there are $5$ types of sundaes, and at most $4$ sundaes of one type are allowed. 75 | 76 | \item Number of ways to get a sum of $25$ when $10$ distinct dice are rolled. 77 | 78 | \item Number of ways to select $300$ chocolate candies from $7$ types of candy if each type comes in boxes of $20$, and at least $1$ but not more than $5$ boxes of each type are chosen. 79 | 80 | \item Number of ways of distributing $30$ distinct objects into $3$ boxes such that no box is empty. 81 | 82 | \item Using generating functions, find the number of $r$-permutations of objects chosen from unlimited supplies of $n$ types of objects. 83 | 84 | \item Number of $r$-digit quaternary sequences (with digits $0$, $1$, $2$, $3$) having an even number of $0$s and an odd number of $1$s. 85 | 86 | \item Write the exponential generating function for the number of arrangements of $k$ objects chosen from $n$ types with at most $4$ objects of each type. 87 | 88 | \item Number of $n$-digit ternary sequences with 89 | \begin{enumerate}[label=(\roman*)] 90 | \item an even number of $0$s. 91 | \item an even number of $0$s and an even number of $1$s. 92 | \item $0$ and $1$ occurring a positive even number of times. 93 | \item at least one $0$ and at least one $1$. 94 | \item the total number of $0$s and $1$s being even. 95 | \item no digit occurs exactly twice. 96 | \end{enumerate} 97 | \end{enumerate} 98 | \end{document} -------------------------------------------------------------------------------- /Programming/MAC 2203/RBTree.py: -------------------------------------------------------------------------------- 1 | from enum import Enum 2 | 3 | class Colour(Enum): 4 | Black = 0 5 | Red = 1 6 | 7 | class Node: 8 | isNil = True 9 | colour = Colour.Black 10 | 11 | def __init__(self, val: int = None, colour: Colour = Colour.Black, left = None, right = None, parent = None): 12 | if val: 13 | self.isNil = False 14 | self.val = val 15 | self.colour = colour 16 | self.left = left 17 | self.right = right 18 | self.parent = parent 19 | 20 | def inorder(self): 21 | if self.isNil: 22 | return [] 23 | else: 24 | return self.left.inorder() + [self.val] + self.right.inorder() 25 | 26 | def structure(self): 27 | if self.isNil: 28 | return "" 29 | else: 30 | return "(" + self.left.structure() + ")<-" + ('R' if self.colour == Colour.Red else 'B') + str(self.val) + "->(" + self.right.structure() + ")" 31 | 32 | def blackHeights(self, h = -1): 33 | if self.isNil: 34 | print(h + 1) 35 | else: 36 | self.left.blackHeights(h + 1 if self.left.colour == Colour.Black else h) 37 | self.right.blackHeights(h + 1 if self.right.colour == Colour.Black else h) 38 | 39 | class RBTree: 40 | Nil = Node() 41 | root = Nil 42 | 43 | def inorder(self): 44 | print(self.root.inorder()) 45 | 46 | def structure(self): 47 | return self.root.structure() 48 | 49 | def insert(self, val): 50 | x = Node(val, colour = Colour.Red, left = self.Nil, right = self.Nil) 51 | y = self.Nil 52 | z = self.root 53 | 54 | while not z.isNil: 55 | y = z 56 | if x.val < z.val: 57 | z = z.left 58 | else: 59 | z = z.right 60 | x.parent = y 61 | 62 | if y.isNil: 63 | self.root = x 64 | elif x.val < y.val: 65 | y.left = x 66 | else: 67 | y.right = x 68 | 69 | self.insertFix(x) 70 | 71 | def insertFix(self, x): 72 | while x.parent.colour == Colour.Red: 73 | if x.parent == x.parent.parent.left: 74 | y = x.parent.parent.right 75 | if y.colour == Colour.Red: 76 | x.parent.colour = Colour.Black 77 | y.colour = Colour.Black 78 | x.parent.parent.colour = Colour.Red 79 | x = x.parent.parent 80 | else: 81 | if x == x.parent.right: 82 | x = x.parent 83 | self.leftRotate(x) 84 | x.parent.colour = Colour.Black 85 | x.parent.parent.colour = Colour.Red 86 | self.rightRotate(x.parent.parent) 87 | else: 88 | y = x.parent.parent.left 89 | if y.colour == Colour.Red: 90 | x.parent.colour = Colour.Black 91 | y.colour = Colour.Black 92 | x.parent.parent.colour = Colour.Red 93 | x = x.parent.parent 94 | else: 95 | if x == x.parent.left: 96 | x = x.parent 97 | self.rightRotate(x) 98 | x.parent.colour = Colour.Black 99 | x.parent.parent.colour = Colour.Red 100 | self.leftRotate(x.parent.parent) 101 | self.root.colour = Colour.Black 102 | 103 | def leftRotate(self, x): 104 | y = x.right 105 | x.right = y.left 106 | if not y.left.isNil: 107 | y.left.parent = x 108 | y.parent = x.parent 109 | if x.parent.isNil: 110 | self.root = y 111 | elif x == x.parent.left: 112 | x.parent.left = y 113 | else: 114 | x.parent.right = y 115 | y.left = x 116 | x.parent = y 117 | 118 | def rightRotate(self, x): 119 | y = x.left 120 | x.left = y.right 121 | if not y.right.isNil: 122 | y.right.parent = x 123 | y.parent = x.parent 124 | if x.parent.isNil: 125 | self.root = y 126 | elif x == x.parent.right: 127 | x.parent.right = y 128 | else: 129 | x.parent.left = y 130 | y.right = x 131 | x.parent = y 132 | 133 | # Prints blackheights as defined along every path from the root to leaves 134 | def blackHeights(self): 135 | self.root.blackHeights() 136 | 137 | 138 | t = RBTree() 139 | for val in [50, 25, 75, 12, 1, 2, 3, 4, 100, 300, 10, 25, 15, 2]: 140 | t.insert(val) 141 | 142 | print(t.structure()) 143 | 144 | print("Blackheights:") 145 | t.blackHeights() -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 10.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx, tikz} 6 | \usepackage{mathtools, physics} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[default,bold]{sourceserifpro} 12 | \usepackage[T1]{fontenc} 13 | \usepackage[]{fouriernc} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\scriptsize\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | 24 | \title[]{Probability and Statistics -- Problem Set 10} 25 | 26 | \DeclareMathOperator{\Prob}{P} 27 | \DeclareMathOperator{\EV}{\mathbb E} 28 | \DeclareMathOperator{\Var}{\mathbb V} 29 | 30 | 31 | \begin{document} 32 | \maketitle 33 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 34 | \item If $X_1, \ldots, X_n$ is an independent sample of size $n$ from a population defined by the pmf $f(x) = \theta^x (1 - \theta)^{1 - x}$, $x = 0, 1$, where $0 < \theta < 1$, find the maximum likelihood estimator for $\theta$. 35 | 36 | \item Find the maximum likelihood estimator for $\theta$ in each of the following cases: 37 | \begin{enumerate} 38 | \item $f(x; \theta) = \theta x^{\theta - 1}$, $0 < x < 1$, where $\theta > 0$. 39 | \item $f(x; \theta) = \theta e^{-\theta x}$, $x > 0$, where $\theta > 0$. 40 | \item $f(x; \theta) = e^{-(x - \theta)}$, $x \ge \theta$, where $\theta$ is any real number. 41 | \item $f(x; \theta) = \frac 1 2 e^{-|x - \theta|}$, where $\theta$ is any real number. 42 | \item $f(x; \theta) = \dfrac {2x} {\theta^2}$, $0 \le x \le \theta$, where $\theta > 0$. 43 | \end{enumerate} 44 | 45 | \item Find the mles of $\theta_1$ and $\theta_2$ in each of the following cases: 46 | \begin{enumerate} 47 | \item $X \sim N(\theta_1, \theta_2)$ 48 | \item $f(x; \theta_1, \theta_2) = \dfrac{1}{\theta_2} e^{- \frac{(x - \theta_1)}{\theta_2}}$, $x > \theta_1$ 49 | \end{enumerate} 50 | where $\theta_1$ is any real number and $\theta_2 > 0$. 51 | 52 | \item Find the mle of $\theta$ in each of the following cases: 53 | \begin{enumerate} 54 | \item $X \sim B(m, \theta)$. 55 | \item $X \sim \mathcal P(\theta)$. 56 | \item $X \sim N(\mu, \theta)$. 57 | \end{enumerate} 58 | 59 | \item Let $X \sim U[0, \theta]$, where $\theta > 0$. Compute the mle for $\theta$. 60 | 61 | \item Show that the sample mean is an unbiased and consistent estimator of population mean. 62 | 63 | \item Show that sample variance is a biased estimator of the population variance. 64 | 65 | \item Show that $Y = \frac{1}{n - 1} \sum_{i = 1}^n (X_i - \overline X)^2$ is an unbiased estimator of population variance. 66 | 67 | \item If $X \sim N(0, \theta)$, show that $Y = \frac 1 n \sum_{i=1}^n X_i^2$ is an unbiased estimator of $\theta$ and has variance $\frac {2\theta^2} n$. 68 | 69 | \item Show that $\overline X$ is an unbiased estimator of $\theta$ if the pdf of $X$ is $f(x; \theta) = \dfrac 1 \theta e^{-\frac x \theta}$, $x > 0$, where $\theta > 0$. Also show that $\overline X$ has variance $\dfrac {\theta^2} n$ and is therefore a consistent estimator. 70 | 71 | \item Let $Y_n$ be an unbiased estimator of $\theta$, such that $V(Y_n) \to 0$ as $n \to \infty$. Then show that $Y_n$ is consistent. 72 | 73 | \item Let $Y_1$ and $Y_2$ be two independent unbiased statistics for $\theta$ such that the variance of $Y_1$ is twice that of $Y_2$. Find the constants $k_1$ and $k_2$ such that $Z = k_1 Y_1 + k_2 Y_2$ is an unbiased statistic for $\theta$ with minimum possible variance for such a linear combination. 74 | 75 | \item Find the same size $n$ such that $\Prob[\overline X - 1 < \mu < \overline X + 1] = 0.9$, given that $X \sim N(\mu, 9)$. 76 | 77 | \item If the observed value of the mean of a sample of size $20$ from a population having distribution $N(\mu, 80)$ is $\overline x = 81.2$, find a $95$ percent confidence interval for the population mean. 78 | 79 | \item If a random sample of size $17$ from a normal distribution $N(\mu, \sigma^2)$ yields $\overline x = 4.7$ and $s^2 = 5.76$, determine a $90$ percent confidence interval for $\mu$. 80 | 81 | \item The observed values of a random sample from a population having distribution $N(8, \sigma^2)$ are $8.6$, $7.9$, $8.3$, $6.4$, $8.4$, $9.8$, $7.2$, $7.8$, and $7.5$. Construct a $90$ percent confidence interval for $\sigma^2$. 82 | 83 | \item A random sample of size $9$ from the distribution $N(\mu, \sigma^2)$ yields $s^2 = 7.63$. Determine a $95$ percent confidence interval for $\sigma^2$. 84 | 85 | \item Find an approximate $95$ percent confidence interval for the mean of a population having variance $100$, if the sample size is $25$. 86 | 87 | \item A random sample of size $15$ from a normal population with unknown mean and variance yields $\overline x = 3.2$ and $s^2 = 4.24$. Determine a $95$ percent confidence interval for $\sigma^2$. 88 | 89 | \item If a sample of size $15$ from a population with distribution $N(\mu, \sigma^2)$ yields values $\sum_{i=1}^{15} X_i = 8.7$ and $\sum_{i = 1}^{15} X_i^2 = 27.3$, obtain a $95$ percent confidence interval for $\sigma^2$. 90 | 91 | \item Suppose that $X \sim N(8, \sigma^2)$, and the observed values of a sample are of size $9$ from this population are $8.6$, $7.9$, $8.3$, $6.4$, $8.4$, $9.8$, $7.2$, $7.8$, and $7.5$. Construct a $90$ percent confidence interval for $\sigma^2$. 92 | 93 | \item Suppose that $X \sim N(\mu, 4)$. If $\overline x = 78.3$ with $n = 25$, obtain a $99$ percent confidence interval for $\mu$. 94 | 95 | \end{enumerate} 96 | 97 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 2.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx} 6 | \usepackage{mathtools} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\thesection.\arabic*.} 10 | 11 | \usepackage[default,bold]{sourceserifpro} 12 | \usepackage[T1]{fontenc} 13 | \usepackage[]{fouriernc} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\scriptsize\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | 24 | \title[]{Probability and Statistics -- Problem Set 2} 25 | 26 | \begin{document} 27 | \maketitle 28 | 29 | 30 | \section{Very Easy} 31 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 32 | \item If $P(A) = 0.5$, $P(A \cap B) = 0.1$, and $P(A \cup B) = 0.8$, what is $P(B)$? 33 | 34 | \item If $P(A) = \frac 1 3$, $P(B) = \frac 2 3$, and $P(AB) = \frac 1 6$, compute 35 | \begin{enumerate} 36 | \item $P(A \cup B)$ 37 | \item $P(\overline A \cup \overline B)$ 38 | \item $P(\overline A \cup B)$. 39 | \end{enumerate} 40 | 41 | \item An integer between $1$ and $n$ (inclusive) is selected at random. Let $A$ be the event that it is even, and let $B$ be the event that it is \textbf{not} divisible by $3$. In each of the following cases, check whether $A$ and $B$ are independent. Guess the answer intuitively before checking it computationally. 42 | \begin{enumerate} 43 | \item $n = 12$. 44 | \item $n = 13$. 45 | \item $n = 14$. 46 | \item $n = 15$. 47 | \item $n = 16$. 48 | \end{enumerate} 49 | \end{enumerate} %Very Easy 50 | 51 | \section{Easy} 52 | \begin{enumerate}[leftmargin=*] 53 | \item $A$ and $B$ are independent events such that $P(A \cup B) = \frac 5 6$ and $P(AB) = \frac 1 4$. Find $P(A)$ and $P(B)$. 54 | 55 | \item Let $p$ and $q$ be distinct prime numbers, and $n$ a positive integer multiple of $pq$. An integer between $1$ and $n$ (inclusive) is selected at random. Let $A$ be the event that \textbf{it is divisible by $p$}, and let $B$ be the event that \textbf{it is not divisible by $q$}. Are $A$ and $B$ independent? 56 | 57 | \item If $P(A) = \frac 1 3$ and $P(B) = \frac 1 4$, show that 58 | \begin{enumerate} 59 | \item $P(AB) \le \frac 1 4$. 60 | \item $\frac 1 3 \le P(A \cup B) \le \frac 7 {12}$ 61 | \end{enumerate} 62 | 63 | \item An evaluation of a small business by an accountant either reveals a problem or does not reveal a problem. The evaluation is either correct or incorrect. The probability that the evaluation is correct is $0.85$. The probability that the evaluation is incorrect and it reveals a problem is $0.1$. If the probability that the evaluation is correct and it does not reveal a problem is $0.25$, what is the probability that the evaluation does not reveal a problem? 64 | 65 | \item If an aircraft is present in a certain area, a radar detects it and generates an alarm signal with probability $0.99$. If an aircraft is not present, the radar generates a (false) alarm with probability $0.1$. Assume that an aircraft is present with probability $0.05$. What is the probability of no aircraft presence and no detection? What is the probability of aircraft presence and detection? What is the probability that an alarm is generated? What is the probability that, if an alarm is generated, an aircraft is present? 66 | \end{enumerate} %Easy 67 | 68 | \section{Normal Difficulty} 69 | \begin{enumerate}[leftmargin=*] 70 | \item If $A$ and $B$ are two events in a sample space, show that $A$ and $B$ are independent if and only if $A$ and $\overline B$ are independent. 71 | 72 | \item Let $p_1, \ldots, p_k$ be $k$ distinct primes, and $n = p_1^{r_1} \times \cdots \times p_k^{r_k}$, where $r_1, \ldots, r_k$ are positive integers. An integer between $1$ and $n$ (inclusive) is selected at random. Let $A_i$ be the event that \textbf{it is not divisible by $p_i$}, $i = 1, \ldots, k$. Find $P(A_1 A_2 \cdots A_k)$, and show that $A_1, \ldots, A_k$ are independent. 73 | 74 | \item In $n$ tosses of a fair coin, let $A$ be the event that there is at least one head and one tail. Let $B$ be the event that there is at most one head. What is the value of $n$ such that $A$ and $B$ are independent? 75 | 76 | \item A fair $4$-sided die is rolled twice. Let $X$ and $Y$ be the outcomes of the first and second rolls, respectively. Determine the conditional probability $P(A \mid B)$, where $A$ is the event that $\max(X, Y) = 3$, and $B$ is the event that $\min(X, Y) = 3$. 77 | 78 | \item A fair $4$-sided die is rolled once. If the result is $1$ or $2$, it is rolled once more, otherwise not. What is the probability that the total of the outcome(s) is at least $4$? 79 | 80 | \item A patient is suspected to have one of three diseases, $A$, $B$, $C$. The population percentages suffering from these diseases are in the ratio $2:1:1$ (and no person has more than one of the three). There is a single test for these three diseases, which turns out to be positive in $25\%$ of the cases of $A$, in $50\%$ of the cases of $B$, and $90\%$ of the cases of $C$. The test is administered to the patient three times. Given that two of them were positive, compute the probability of the patient having each of these diseases. 81 | 82 | \item Urn $A$ contains $5$ red and $3$ blue marbles. Urn $B$ contains $4$ red and $6$ blue marbles. One marble chosen at random is transferred from $A$ to $B$. Then, one is chosen at random from $B$ and transferred to $A$. Finally, one marble is drawn at random from $A$. 83 | \begin{enumerate} 84 | \item What is the probability that the marble drawn from $A$ in the third step is red? 85 | \item Given that this marble is red, what is the probability that the marble taken from $A$ in the first step is also red? 86 | \end{enumerate} 87 | \end{enumerate} %Normal 88 | 89 | \section{Seemingly Difficult} 90 | \begin{enumerate}[leftmargin=*] 91 | \item An integer between $1$ and $n$ (inclusive) is selected at random. Let $A$ be the event that it is even, and let $B$ be the event that it is \textbf{not} divisible by $3$. For which values of $n$ will $A$ and $B$ be independent? 92 | 93 | \item A certain family of bacteria is such that every ten minutes, each bacterium either divides into two, with probability $\frac 2 3$, or dies. If initially there is a single bacterium of this family, what is the probability that its lineage never dies out? 94 | \end{enumerate} %Difficult 95 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 3.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx, tikz} 6 | \usepackage{mathtools, physics} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[]{fouriernc} 12 | \usepackage[default,bold]{sourceserifpro} 13 | \usepackage[T1]{fontenc} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\scriptsize\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | 24 | \title[]{Probability and Statistics -- Problem Set 3} 25 | 26 | \DeclareMathOperator{\Prob}{P} 27 | \DeclareMathOperator{\EV}{\mathbb E} 28 | \DeclareMathOperator{\Var}{\mathbb V} 29 | 30 | 31 | \begin{document} 32 | \maketitle 33 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 34 | \item $X \sim U[a,b]$, $a < 1 < 2 < b$. If $\Prob[X < 1] = 2\Prob[X > 2]$, and $5\Prob[X > 1] = 3\Prob[X < 2]$, find the values of $a$ and $b$. 35 | 36 | A fair coin is tossed three times. A player wins $\$1$ if the first toss results in a head, but loses $\$1$ if the first toss results in a tail. Similarly, the player wins $\$2$ if the second toss results in a head, but loses $\$2$ if it results in a tail, and wins or loses $\$3$ according to the result of the third toss. Let $X$ be the total winnings after the three tosses (possibly a negative value if losses are incurred). 37 | \begin{enumerate} 38 | \item Compute the probability mass function. 39 | \item Compute the cumulative distribution function. 40 | \item What is the most likely value of $X$? 41 | \end{enumerate} 42 | 43 | \item A contestant on a quiz show is presented with two questions, Questions 1 and 2, which she is to attempt to answer in some order she chooses. If she decides to try Question $i$ first, then she will be allowed to go on to question $j$, $j \neq i$, only if her answer to question $i$ is correct. If her initial answer is incorrect, she is not allowed to answer the other question. 44 | 45 | If she is $60$ percent certain of answering Question 1, worth $200$ dollars, correctly and she is $80$ percent certain of answering Question 2, worth 100 dollars, correctly, then which question should she attempt to answer first so as to maximise her expected winnings? Assume that the events $E_i$, $i = 1, 2,$ that she knows the answer to question $i$ are independent 46 | events. 47 | 48 | \item The diameter of an electric cable $X$ is assumed to be a continuous random variable with pdf $f(x) = kx (1- x)$, $0 \leq x \leq 1$. 49 | \begin{enumerate} 50 | \item Find the value of $k$. 51 | \item Find the cdf of $X$. 52 | \item Determine $b$ such that $P(X < b) = 2 P(X \geq b)$. 53 | \end{enumerate} 54 | 55 | \item A student takes a multiple choice test consisting of three problems. The first question has $3$ possible answers, the second has $5$ possible answers, and the third question has $4$ possible answers. The student chooses at random one answer as the right one from each of the three problems. Let $X$ be the number of right answers. Find $\EV[X]$ and $\EV[X^2] - \EV[X]^2$. 56 | 57 | \item $X$ is a discrete random variable taking values $0, 1, 2, \ldots$, whose pmf $f(x)$ is such that $\dfrac{f(x_1)}{f(x_2)} = r^{x_1 - x_2}$ for all $x_1, x_2 = 0, 1, 2, \ldots$ (where $r \in (0, 1)$ is a constant). Determine $f(x)$ and compute $\EV[X]$. 58 | 59 | \item Let $X$ be the random variable with pdf $f(x)$ whose graph is shown below (where $a < b < c$ are three real numbers).\\ 60 | \includegraphics[scale=0.6]{Set3Graph.pdf} \\ 61 | Find $h$, and compute $\EV[X]$. 62 | 63 | \item Show that the function 64 | \begin{equation*} 65 | f(x) = \dfrac{\lambda^x}{x!}e^{-\lambda}, x = 0, 1, 2, \ldots 66 | \end{equation*} 67 | (where $\lambda > 0$ is a constant) is a valid pmf, and compute $\EV[X]$ and $\Var[X]$. 68 | 69 | \item $X$ is a discrete random variable with pmf $f(x) = \dfrac{6}{\pi^2 x^2}$, $x = 1, 2, \ldots$. 70 | \begin{enumerate} 71 | \item Compute $\Prob[X \ge 3]$ 72 | \item Compute $\Prob[10 \le X \le 15]$ 73 | \item Show that $\EV[X]$ does not exist. 74 | \end{enumerate} 75 | 76 | \item $X$ is a continuous random variable with pdf $f(x) = \dfrac 6 {\pi^2} \pqty{\dfrac 1 {\lfloor x \rfloor}}^2$, $x > 1$. 77 | \begin{enumerate} 78 | \item Compute $\Prob[X \ge 3]$ 79 | \item Compute $\Prob[10 \le X \le 15]$ 80 | \item Show that $\EV[X]$ does not exist. 81 | \end{enumerate} 82 | 83 | \item $X$ has pdf $f(x) = \dfrac{x^2}{3}$, $-1 \le x \le 2$. Compute 84 | \begin{enumerate} 85 | \item $\Prob \bqty{X < \frac 3 2}$ 86 | \item $\Prob[|X| > 1]$ 87 | \item $\Prob \bqty{X < \frac 3 2 \mid X > \frac 1 2}$ 88 | \item $\Prob \bqty{X < \frac 3 2 \mid |X| > 1}$ 89 | \item $\Prob \bqty{|X| < 1 \mid X < \frac 3 2}$. 90 | \end{enumerate} 91 | 92 | \item The amount of time in hours that a computer functions before breaking down is a continuous random variable with probability density function given by $f(x) = k e^{\frac{-x}{100}}$, $x \ge 0$. 93 | 94 | What is the probability that 95 | \begin{enumerate} 96 | \item the computer will function between $50$ and $150$ hours before breaking down? 97 | \item it will function for fewer than $100$ hours? 98 | \end{enumerate} 99 | 100 | \item An urn initially has {\color{red} $1$ red} and {\color{blue} $1$ blue} marble. A marble is drawn at random from the urn, and if it is {\color{blue} blue}, it is put back and {\color{red} one red} marble is added to the urn. This is continued until a {\color{red} red} marble is drawn. Let $X$ denote the total number of draws required to obtain a {\color{red} red} marble. Determine the probability distribution of $X$, and find its mean and variance. 101 | {\scriptsize\textbf{Hint}: Compute $\EV[X + 1]$ and $\EV[X^2 - 1]$.} 102 | 103 | \item The \emph{median} of a random variable $X$ is the point $c$ such that $\Prob[X \le c] = \Prob[X \ge c] = \frac 1 2$. Compute the median of $X$ with pdf $f(x)$ in each of the following cases. 104 | \begin{enumerate}[itemsep=1em] 105 | \item $f(x) = \dfrac 1 {b - a}$, $a < x < b$. 106 | \item $f(x) = a e^{-ax}$, $x > 0$ [where $a > 0$ is a constant]. 107 | \item $f(x) = \dfrac{1}{\pi(1 + x^2)}$ [Note that this distribution has no mean, but has a median]. 108 | \end{enumerate} 109 | 110 | \item Suppose that if you are $s$ minutes early for an appointment, then you incur the cost $cs$, and if you are $s$ minutes late, then you incur the cost $ks$. Suppose also that the travel time from where you presently are to the location of your appointment is a continuous random variable having probability density function $f$. Determine the time at which you should depart if you want to minimise your expected cost. 111 | \end{enumerate} 112 | \end{document} -------------------------------------------------------------------------------- /Notes/Estimation of Parameters/Estimation of Parameters.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames, a5paper]{article} 2 | \usepackage[top = 20mm, bottom = 15mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{natbib} 5 | 6 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 7 | \setcounter{tocdepth}{3} 8 | \usepackage{physics} 9 | 10 | \usepackage{mathtools} 11 | \usepackage{xspace} 12 | \usepackage{enumitem} 13 | 14 | 15 | \usepackage{tocloft} 16 | \renewcommand{\cftdot}{} 17 | 18 | \usepackage{tikz} 19 | \usepackage{hyperref} 20 | \hypersetup{colorlinks, linkcolor = [RGB]{66, 128, 128}, urlcolor = red, linktocpage = true} 21 | \usepackage{bookmark} 22 | \bookmarksetup{color = [RGB]{66, 128, 128}} 23 | 24 | \usepackage{hhline, array, multirow} 25 | 26 | \usepackage[intoc]{nomencl} 27 | \makenomenclature 28 | 29 | \usepackage[toc, page]{appendix} 30 | 31 | \usepackage{newpxmath} 32 | \usepackage{charter} 33 | \usepackage[T1]{fontenc} 34 | 35 | \usepackage{scrlayer-scrpage} 36 | \ohead{\color{blue!35!black} \scshape VM} 37 | \cfoot*{\pagemark} 38 | 39 | \newtheorem{Theorem}{Theorem}[section] 40 | \newtheorem{Lemma}[Theorem]{Lemma} 41 | \newtheorem{Corollary}[Theorem]{Corollary} 42 | 43 | \theoremstyle{definition} 44 | \newtheorem{Definition}[Theorem]{Definition} 45 | \newtheorem*{Definition*}{Definition} 46 | \newtheorem{Example}[Theorem]{Example} 47 | \newtheorem*{Example*}{Example} 48 | \newtheorem{Exercise}{Exercise}[section] 49 | 50 | \theoremstyle{remark} 51 | \newtheorem*{Remark*}{Remark} 52 | \newtheorem*{Solution*}{Solution} 53 | \newtheorem*{Note*}{Note} 54 | 55 | \DeclareMathOperator{\ord}{o} 56 | \newcommand{\Mod}[1]{\ (\mathrm{mod}\ #1)} 57 | \DeclareMathOperator{\lcm}{lcm} 58 | 59 | \let\Im\relax 60 | \DeclareMathOperator{\Im}{Im} 61 | \newcommand{\id}{\mathrm{id}} 62 | \renewcommand{\th}{\textsuperscript{th}\xspace} 63 | 64 | \newlist{subquests}{enumerate}{2} 65 | \setlist[subquests, 1]{label = (\alph*)} 66 | \setlist[subquests, 2]{label = \roman*.} 67 | 68 | \begin{document} 69 | 70 | \title{\textbf{Estimation of Parameters}} 71 | 72 | \date{} 73 | \maketitle 74 | 75 | %\begingroup 76 | %\let\clearpage\relax 77 | %\tableofcontents 78 | %\endgroup 79 | 80 | \section{Maximum Likelihood Estimation}\label{sec:MLE} 81 | 82 | Let $X$ be a random variable having pdf or pmf $f(x; \theta)$, where $\theta$ is an unknown parameter. The \emph{likelihood function} of $\theta$, corresponding to a sample of size $n$, is 83 | \begin{equation*} 84 | L(\theta) = f(x_1; \theta) f(x_2; \theta) \cdots f(x_n; \theta). 85 | \end{equation*} 86 | The value of $\theta = T(x_1, \ldots, x_n)$ (say) for which $L(\theta)$ is maximum is the \emph{maximum likelihood estimate} of $\theta$. The corresponding statistic $\hat\theta = T(X_1, \ldots, X_n)$ is the \emph{maximum likelihood estimator} (mle) of $\theta$. 87 | 88 | Where applicable, we maximise $L(\theta)$ by solving the equation $\dfrac{d}{d\theta} L(\theta) = 0$ for $\theta$. As $L(\theta)$ is the product of $f(x_1; \theta), \ldots, f(x_n; \theta)$, it is more often convenient to maximise the \emph{log-likelihood function} $\log L(\theta) = \sum_{i=1}^{n} \log f(x_i; \theta)$ -- which yields the same estimator $\hat\theta$, since $\log$ is an increasing function. 89 | 90 | \begin{Example} 91 | Let $X \sim \mathcal E(\theta)$ (an exponential distribution with parameter $\theta$). Then the pdf of $X$ is $f(x; \theta) = \theta e^{-\theta x}$, $x \ge 0$, where $\theta > 0$. The likelihood function (assuming a sample of size $n$) will be 92 | \begin{equation*} 93 | L(\theta) = \theta^n e^{-\theta \sum_{i=1}^n x_i} 94 | \end{equation*} 95 | and therefore the log-likelihood function will be 96 | \begin{equation*} 97 | \log L(\theta) = n \log \theta - \theta \sum_{i = 1}^n x_i. 98 | \end{equation*} 99 | Differentiating this with respect to $\theta$ and equating to zero, we have 100 | \begin{equation*} 101 | \frac n \theta - \sum_{i = 1}^n x_i = 0 102 | \end{equation*} 103 | which implies that 104 | \begin{equation*} 105 | \theta = \dfrac n {\sum_{i = 1}^n x_i} = \dfrac{1}{\overline x}. 106 | \end{equation*} 107 | Thus, the MLE of $\theta$ is 108 | \begin{equation*} 109 | \hat \theta = \dfrac 1 {\overline X}. 110 | \end{equation*} 111 | \end{Example} 112 | 113 | \begin{Example} 114 | Let $X$ be a random variable with pdf $f(x; \theta) = (1 + \theta) x^{\theta}$, $0 < x < 1$, where $\theta > 0$. The likelihood function will be 115 | \begin{equation*} 116 | L(\theta) = (1 + \theta)^n x_1^\theta \cdots x_n^\theta 117 | \end{equation*} 118 | and hence 119 | \begin{equation*} 120 | \log L(\theta) = n \log(1 + \theta) + \theta \sum_{i = 1}^n \log x_i. 121 | \end{equation*} 122 | Equating its derivative with respect to $\theta$ to zero, we have 123 | \begin{equation*} 124 | \dfrac{n}{1 + \theta} + \sum_{i = 1}^n \log x_i = 0 125 | \end{equation*} 126 | which implies that the MLE of $\theta$ is 127 | \begin{align*} 128 | \hat\theta = \dfrac{n}{\sum\limits_{i = 1}^n \log \frac 1 {X_i}} - 1. 129 | \end{align*} 130 | \end{Example} 131 | 132 | \section{Confidence Intervals}\label{sec:CI} 133 | 134 | \begin{Definition} 135 | If $(X_1, \ldots, X_n)$ is a random sample of size $n$, then the \emph{sample mean} is defined as 136 | \begin{equation*} 137 | \overline X = \dfrac 1 n \sum_{i = 1}^n X_i 138 | \end{equation*} 139 | and the \emph{sample variance} is defined as 140 | \begin{equation*} 141 | S^2 = \dfrac 1 n \sum_{i = 1}^n \pqty{X_i - \overline X}^2. 142 | \end{equation*} 143 | \end{Definition} 144 | 145 | \begin{Theorem} 146 | Suppose that $X \sim N(\mu, \sigma^2)$. Then the following hold: 147 | \begin{enumerate} 148 | \item $\dfrac{\overline X - \mu}{\sigma / \sqrt n} \sim N(0, 1)$. 149 | \item $\dfrac{\overline X - \mu}{S/\sqrt{n - 1}} \sim t_{n-1}$. 150 | \item $\dfrac 1 {\sigma^2} \displaystyle\sum_{i=1}^n (X_i - \mu)^2 \sim \chi^2_n$. 151 | \item $\dfrac{nS^2}{\sigma^2} \sim \chi^2_{n - 1}$. 152 | \end{enumerate} 153 | \end{Theorem} 154 | 155 | Confidence intervals for the mean and variance of a normal population, under different conditions, are as described in the table given below. 156 | \begin{center} 157 | \def\arraystretch{2.3} 158 | \begin{tabular}{|c|c|>{\setlength{\baselineskip}{2.2\baselineskip}\centering\arraybackslash}p{0.45\textwidth}|} 159 | \hline 160 | \textbf{Parameter} & \textbf{Condition} & \textbf{Interval} \\ 161 | \hhline{|===|} 162 | \multirow{2}{*}{$\mu$} & $\sigma^2$ known & 163 | $\pqty{ \overline x - \dfrac{a\sigma}{\sqrt n} , \overline x + \dfrac{a\sigma}{\sqrt n} }$ where $P[-a < Z < a] = p$, $Z \sim N(0, 1)$ \\ 164 | \cline{2-3} 165 | & $\sigma^2$ unknown & 166 | $\pqty{ \overline x - \dfrac{bS}{\sqrt {n-1}} , \overline x + \dfrac{bS}{\sqrt {n-1}} }$ where $P[-b < T < b] = p$, $T \sim t_{n-1}$ \\ 167 | \hline 168 | \multirow{2}{*}{$\sigma^2$} & $\mu$ known & 169 | $\pqty{\dfrac 1 b \sum\limits_{i=1}^n (x_i - \mu)^2, \dfrac 1 a \sum\limits_{i=1}^n (x_i - \mu)^2}$ where $P[Z < a] = P[Z > b] = \frac {1-p} 2$, $Z \sim \chi^2_n$ \\ 170 | \cline{2-3} 171 | & $\mu$ unknown & $\pqty{ \dfrac{ns^2}{b}, \dfrac{ns^2}{a}}$ where $P[Z < a] = P[Z > b] = \frac {1-p} 2$, $Z \sim \chi^2_{n-1}$ \\ 172 | \hline 173 | \end{tabular} 174 | \end{center} 175 | 176 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 5.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx, tikz} 6 | \usepackage{mathtools, physics} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\arabic*.} 10 | 11 | \usepackage[default,bold]{sourceserifpro} 12 | \usepackage[T1]{fontenc} 13 | \usepackage[]{fouriernc} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\scriptsize\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | 24 | \title[]{Probability and Statistics -- Problem Set 5} 25 | 26 | \DeclareMathOperator{\Prob}{P} 27 | \DeclareMathOperator{\EV}{\mathbb E} 28 | \DeclareMathOperator{\Var}{\mathbb V} 29 | 30 | 31 | \begin{document} 32 | \maketitle 33 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 34 | \item Suppose that the number of items produced in a factory during one week is a random variable with mean $500$ and variance $100$. 35 | \begin{enumerate} 36 | \item What can be said about the probability that this week's production will be at least $1000$? 37 | \item What can be said about the probability that this week's production will be between $400$ and $600$? 38 | \end{enumerate} 39 | 40 | \item If the time required to complete a task is a random variable with mean $20$ minutes and standard deviation $3$ minutes, find the smallest time frame such that the probability that the task will be completed within the time frame is at least $0.75$. 41 | 42 | \item If $X \sim B(6, p)$, and $\Prob[X = 2] = 9 \Prob[X = 4]$, then find $\Prob[X \le 3]$. 43 | 44 | \item An urn contains $4$ red, $4$ green, and $2$ blue marbles. One marble is drawn at random from the urn, observed, and placed back. Find the probability that in $8$ such draws 45 | \begin{enumerate} 46 | \item exactly $3$ red marbles are drawn. 47 | \item not more than $3$ red marbles are drawn. 48 | \item at least $3$ red marbles are drawn. 49 | \item not all marbles drawn are red. 50 | \end{enumerate} 51 | 52 | \item The number of visitors to a webpage per minute follows a Poisson distribution. If the average number of visitors per minute is $4$, what is the probability that the webpage receives 53 | \begin{enumerate}[label=(\roman*)] 54 | \item exactly four visitors in one minute? 55 | \item at least two visitors in one minute? 56 | \end{enumerate} 57 | 58 | \item If $X$ is a Poisson variate such that $3 \Prob[X = 2] = 2\Prob[X = 1]$, then what is $E[X]$? 59 | 60 | \item It is observed in a communication channel that $90\%$ of the messages sent are received without any error. Find the probability that among $18$ messages sent through the channel 61 | \begin{enumerate}[label=(\roman*)] 62 | \item at least $16$ are received without any error 63 | \item at most $14$ are received without any error. 64 | \end{enumerate} 65 | 66 | \item In a certain factory producing blades, there is a small probability of $\frac{1}{500}$ for any blade to be defective. The blades are supplied in packets of $10$. Calculate the approximate number of packets containing 67 | \begin{enumerate}[label=(\roman*)] 68 | \item no defective 69 | \item one defective 70 | \item two defective 71 | \item at least two defective 72 | \end{enumerate} 73 | blades in a consignment of $10000$ packets. 74 | 75 | \item One per thousand of a population is subject to certain kinds of accidents each year. Given that an insurance company has insured $5,000$ persons from the population, find the probability that at most $2$ of them will incur this accident. 76 | 77 | \item An airline company, having observed that $5\%$ of the persons making reservations on a flight do not show up for the flight, sells $100$ seats on a plane that has $95$ seats. What is the probability that there will be a seat available for every person who shows up for the flight? 78 | 79 | \item If $X$ is a binomial variate with mean $3$, such that $\Prob[X = 3] = 2\Prob[X = 2]$, find $V[X]$. 80 | 81 | \item $X$ is a Poisson variate and the probability that $X$ is even is twice the probability that it is odd. Determine $\Prob[X = 0]$. 82 | 83 | \item If $X$ is exponentially distributed with $E[X] = 2$, then find the value of $a$ such that $\Prob[X \le a] = \Prob[X \ge a]$.\\ 84 | {\small\color{blue!40!black} 85 | Note: This point $a$ is the \emph{median} of $X$. 86 | } 87 | 88 | \item Suppose that the duration in minutes of a phone call follows an exponential distribution with mean $5$ minutes. 89 | \begin{enumerate} 90 | \item Find the probability that the duration of a particular call 91 | \begin{enumerate} 92 | \item will exceed $5$ minutes 93 | \item will be between $5$ and $6$ minutes 94 | \item will be less than $3$ minutes 95 | \item will be less than $6$ minutes, given that it was greater than $3$ minutes. 96 | \end{enumerate} 97 | \item Suppose exactly $100$ such phone calls are received every day. Find the probability that on a given day 98 | \begin{enumerate} 99 | \item every phone call lasted longer than $8$ minutes 100 | \item at least $4$ phone calls lasted longer than $8$ minutes. 101 | \end{enumerate} 102 | \item Find the probability that in one week, there are at least $3$ days on each which there were at least $4$ phone calls that lasted longer than $8$ minutes. 103 | \end{enumerate} 104 | 105 | \item A random variable $X$ follows an exponential distribution with parameter $\alpha = 3$. Compute 106 | \begin{enumerate} 107 | \item $\Prob[2X > 1]$. 108 | \item $\Prob[X > s + t \mid X > t]$, where $s, t > 0$. 109 | \end{enumerate} 110 | 111 | \item The daily consumption of milk in a city, in excess of 20,000 gallons, is distributed as a Gamma variate with parameters $\alpha = \frac 1 {10000}$ and $r = 2$. The city has a daily stock of $30000$ gallons. What is the probability that the stock is insufficient on a particular day? 112 | 113 | \item If $K \sim U[0, 5]$, then what is the probability that the roots of the equation $4x^2 + 4Kx + K + 2 = 0$ are real? 114 | 115 | \item A book of $200$ pages contains $100$ misprints distributed randomly throughout its pages. If one page is selected at random and examined, what is the probability that it contains 116 | \begin{enumerate} 117 | \item no misprints? 118 | \item at least $2$ misprints? 119 | \end{enumerate} 120 | 121 | \item Suppose that the chances of a traffic accident in a street of a city in a day is $0.001$. In how many days in a year can we expect 122 | \begin{enumerate} 123 | \item no accidents? 124 | \item at least one accident, if there are $100$ such streets in a city? 125 | \end{enumerate} 126 | If there are $5$ such cities in a state, what is the probability that at least one city will have at least one accident in a day? 127 | 128 | \item Buses start from a certain station at intervals of $45$ minutes. What is the probability that a person reaching the station at a random point in time will have to wait for at least $30$ minutes? 129 | 130 | \item In a bombing action, there is a $50\%$ chance that a bomb will hit a target. Two direct hits are needed to destroy the target completely. Estimate the minimum number of bombs to be dropped to give $90\%$ or better chance of completely destroying the target. 131 | \end{enumerate} 132 | \end{document} -------------------------------------------------------------------------------- /Problem Sets/Probability and Statistics/Set 1.tex: -------------------------------------------------------------------------------- 1 | \documentclass[svgnames]{amsart} 2 | \usepackage[paperwidth=6in, paperheight=8in, top = 20mm, bottom = 18mm, left=10mm, right = 10mm]{geometry} 3 | 4 | \usepackage{amsmath, amsfonts, amssymb, amsthm} 5 | \usepackage{graphicx} 6 | \usepackage{mathtools} 7 | 8 | \usepackage{enumitem} 9 | \setlist[enumerate,1]{label=\thesection.\arabic*.} 10 | 11 | \usepackage[default,bold]{sourceserifpro} 12 | \usepackage[T1]{fontenc} 13 | \usepackage[]{fouriernc} 14 | 15 | \usepackage{xcolor} 16 | \usepackage{scrlayer-scrpage} 17 | \ohead{\color{blue!35!black} \scshape VM} 18 | \cfoot*{\scriptsize\pagemark} 19 | 20 | \renewcommand{\th}{\textsuperscript{th}} 21 | 22 | \setlength{\parindent}{0pt} 23 | 24 | \title[]{Probability and Statistics -- Problem Set 1} 25 | 26 | \begin{document} 27 | \maketitle 28 | 29 | \section{Very Easy} 30 | \begin{enumerate}[leftmargin=*, itemsep=0.3em] 31 | \item A box contains $10$ paper slips, labelled $1, 2, \ldots, 10$. Find the probability that one slip drawn at random contains: 32 | \begin{enumerate} 33 | \item the number $9$. 34 | \item an even number. 35 | \item an even number or an odd number. 36 | \item an even number or a prime number. 37 | \end{enumerate} 38 | 39 | \item A fair coin is tossed twice. Find the probability that 40 | \begin{enumerate} 41 | \item A head is obtained on the first toss. 42 | \item A head is obtained on the first toss and a tail on the second. 43 | \item A head is obtained on at least one of the two tosses. 44 | \end{enumerate} 45 | 46 | \item A fair, six-sided die is rolled. Find the probability that the outcome is 47 | \begin{enumerate} 48 | \item $2$ 49 | \item an odd number. 50 | \item an odd number or an even number. 51 | \item an odd number or a composite number. 52 | \end{enumerate} 53 | \end{enumerate} %Very Easy 54 | 55 | \section{Easy} 56 | \begin{enumerate}[leftmargin=*] 57 | \item A box contains $55$ paper slips -- one labelled $1$, two labelled $2$, \ldots, ten labelled $10$ (i.e., $k$ slips labelled $k$, for each $k = 1, \ldots, 10$). Find the probability that one slip drawn at random contains: 58 | \begin{enumerate} 59 | \item the number $9$. 60 | \item an even number. 61 | \item an even number or an odd number. 62 | \item an even number or a prime number. 63 | \end{enumerate} 64 | 65 | \item A coin with probability $1/3$ for heads and $2/3$ for tails is tossed twice. Find the probability that 66 | \begin{enumerate} 67 | \item A head is obtained on the first toss. 68 | \item A head is obtained on the first toss and a tail on the second. 69 | \item A head is obtained on at least one of the two tosses. 70 | \end{enumerate} 71 | 72 | \item A six-sided die is designed in such a way that the probability of occurrence of each face is proportional to the number on that face. Find the probability that the outcome, when the die is rolled once, is 73 | \begin{enumerate} 74 | \item $2$ 75 | \item an odd number. 76 | \item an odd number or an even number. 77 | \item an odd number or a composite number. 78 | \end{enumerate} 79 | 80 | \item Let $m$ and $n$ denote the two outcomes when two fair dice are rolled. Find the probability that 81 | \begin{enumerate} 82 | \item $m = 4$ or $n = 4$. 83 | \item $\max(m, n) = 4$. 84 | \item $\max(m, n) > 4$. 85 | \end{enumerate} 86 | 87 | \item Three marbles are drawn simultaneously at random from a box containing $2$ red, $3$ green, and $5$ blue marbles. Find the probability that 88 | \begin{enumerate} 89 | \item all three are green. 90 | \item all three are blue. 91 | \item all three are red. 92 | \item at least one is red. 93 | \item each one is green or blue. 94 | \item one is red and two are blue. 95 | \end{enumerate} 96 | 97 | \item A box of $100$ lightbulbs manufactured in a factory has $10$ defective lightbulbs. An inspector tests $5$ lightbulbs selected randomly from the box. What is the probability that a defective one will be found? 98 | 99 | \item A group of $2n$ boys and $2n$ girls is randomly divided into two equal groups. What is the probability that each group has the same number of boys and girls? 100 | 101 | \item A box contains $n$ paper slips, labelled $1, 2, \ldots, n$. Find the probability that two slips drawn at random contain consecutive numbers, if they are drawn one after the other 102 | \begin{enumerate} 103 | \item without replacement. 104 | \item with replacement. 105 | \end{enumerate} 106 | \end{enumerate} %Easy 107 | 108 | \section{Normal Difficulty} 109 | \begin{enumerate}[leftmargin=*] 110 | \item A box contains $10$ paper slips, labelled $1, \ldots, 10$. Slips are drawn at random without replacement, until $9$ is obtained. Find the probability that $9$ is obtained 111 | \begin{enumerate} 112 | \item in the $n$\th draw (for each $n = 1, \ldots, 10$). 113 | \item after the $n$\th draw (for each $n = 1, \ldots, 9$). {\scriptsize Note: Not necessarily \emph{immediately} after it.} 114 | \item after $10$ is obtained. 115 | \item immediately after $10$ is obtained. 116 | \item immediately before or after $10$ is obtained. 117 | \end{enumerate} 118 | 119 | \item A coin with probability $1/3$ for heads and $2/3$ for tails is tossed until a head is obtained. Find the probability that 120 | \begin{enumerate} 121 | \item exactly $n$ tosses are required ($n = 1, 2, \ldots$). 122 | \item the number of tosses required is even. 123 | \item at least $n$ tosses are required. 124 | \end{enumerate} 125 | 126 | \item A fair, six-sided die is rolled until the same face is obtained twice in succession. Find the probability that 127 | \begin{enumerate} 128 | \item exactly $n$ rolls are required ($n = 2, 3, \ldots$). 129 | \item $2$ is obtained on the last two rolls (regardless of number of rolls required). 130 | \item $2$ is not obtained on any roll. 131 | \item $2$ is obtained on the last two rolls, but not before. 132 | \end{enumerate} 133 | 134 | \item Let $S$ be a set of $n$ elements, and $\mathcal P(S)$ its power set -- the collection of all subsets of $S$. Let $A$ be a subset of $S$ picked at random from $\mathcal P(S)$. 135 | \begin{enumerate} 136 | \item What is the probability that $A$ has $m$ elements ($0 \le m \le n$)? 137 | \item If $B$ is a given subset of $S$, what is the probability that $A = B$? 138 | \end{enumerate} 139 | 140 | \item Let $S = \{s_1, s_2, \ldots, s_n\}$ be a set of $n$ elements. Construct a random subset $A$ of $S$ as follows: For each $i = 1, \ldots, n$, toss a fair coin and on heads, include the element $s_i$ in $A$, and on tails, exclude $s_i$ from $A$. 141 | \begin{enumerate} 142 | \item What is the probability that $A$ has $m$ elements ($0 \le m \le n$)? 143 | \item If $B$ is a given subset of $S$, what is the probability that $A = B$? 144 | \end{enumerate} 145 | 146 | \item Let $S = \{s_1, s_2, \ldots, s_n\}$ be a set of $n$ elements, and consider a coin weighted such that heads occur twice as often as tails. Construct a random subset $A$ of $S$ as follows: For each $i = 1, \ldots, n$, toss the coin and on heads, include the element $s_i$ in $A$, and on tails, exclude $s_i$ from $A$. 147 | \begin{enumerate} 148 | \item What is the probability that $A$ has $m$ elements ($0 \le m \le n$)? 149 | \item If $B$ is a given subset of $S$, what is the probability that $A = B$? 150 | \end{enumerate} 151 | 152 | \end{enumerate} %Normal 153 | 154 | \section{Seemingly Difficult} 155 | \begin{enumerate}[leftmargin=*] 156 | \item A box contains $n$ paper slips labelled $1, \ldots, n$ ($n \ge 9$). Slips are drawn at random with replacement, until $9$ is obtained. What is the minimum value of $n$ such that the probability that more than $10$ draws are required is at least $0.5$? {\scriptsize (Take your time to understand this question correctly.)} 157 | 158 | \item A coin with probability $1/3$ for heads and $2/3$ for tails is tossed $n$ times. What is the minimum value of $n$ such that the probability that at least $2$ heads are obtained is at least $0.5$? 159 | 160 | \item A fair, six-sided die is rolled until the total of all outcome $2$ is obtained a total of $n$ times (not necessarily in consecutive throws). Find the probability that exactly $m$ throws are required ($m = 1, 2, \ldots$). 161 | \end{enumerate} %Difficult 162 | \end{document} -------------------------------------------------------------------------------- /Notes/Applied Graph Theory/Images/BullGraph.ipe: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 0 0 m 9 | -1 0.333 l 10 | -1 -0.333 l 11 | h 12 | 13 | 14 | 15 | 16 | 0 0 m 17 | -1 0.333 l 18 | -1 -0.333 l 19 | h 20 | 21 | 22 | 23 | 24 | 0 0 m 25 | -1 0.333 l 26 | -0.8 0 l 27 | -1 -0.333 l 28 | h 29 | 30 | 31 | 32 | 33 | 0 0 m 34 | -1 0.333 l 35 | -0.8 0 l 36 | -1 -0.333 l 37 | h 38 | 39 | 40 | 41 | 42 | 0.6 0 0 0.6 0 0 e 43 | 0.4 0 0 0.4 0 0 e 44 | 45 | 46 | 47 | 48 | 0.6 0 0 0.6 0 0 e 49 | 50 | 51 | 52 | 53 | 54 | 0.5 0 0 0.5 0 0 e 55 | 56 | 57 | 0.6 0 0 0.6 0 0 e 58 | 0.4 0 0 0.4 0 0 e 59 | 60 | 61 | 62 | 63 | 64 | -0.6 -0.6 m 65 | 0.6 -0.6 l 66 | 0.6 0.6 l 67 | -0.6 0.6 l 68 | h 69 | -0.4 -0.4 m 70 | 0.4 -0.4 l 71 | 0.4 0.4 l 72 | -0.4 0.4 l 73 | h 74 | 75 | 76 | 77 | 78 | -0.6 -0.6 m 79 | 0.6 -0.6 l 80 | 0.6 0.6 l 81 | -0.6 0.6 l 82 | h 83 | 84 | 85 | 86 | 87 | 88 | -0.5 -0.5 m 89 | 0.5 -0.5 l 90 | 0.5 0.5 l 91 | -0.5 0.5 l 92 | h 93 | 94 | 95 | -0.6 -0.6 m 96 | 0.6 -0.6 l 97 | 0.6 0.6 l 98 | -0.6 0.6 l 99 | h 100 | -0.4 -0.4 m 101 | 0.4 -0.4 l 102 | 0.4 0.4 l 103 | -0.4 0.4 l 104 | h 105 | 106 | 107 | 108 | 109 | 110 | 111 | -0.43 -0.57 m 112 | 0.57 0.43 l 113 | 0.43 0.57 l 114 | -0.57 -0.43 l 115 | h 116 | 117 | 118 | -0.43 0.57 m 119 | 0.57 -0.43 l 120 | 0.43 -0.57 l 121 | -0.57 0.43 l 122 | h 123 | 124 | 125 | 126 | 127 | 128 | 0 0 m 129 | -1 0.333 l 130 | -1 -0.333 l 131 | h 132 | 133 | 134 | 135 | 136 | 0 0 m 137 | -1 0.333 l 138 | -0.8 0 l 139 | -1 -0.333 l 140 | h 141 | 142 | 143 | 144 | 145 | 0 0 m 146 | -1 0.333 l 147 | -0.8 0 l 148 | -1 -0.333 l 149 | h 150 | 151 | 152 | 153 | 154 | -1 0.333 m 155 | 0 0 l 156 | -1 -0.333 l 157 | 158 | 159 | 160 | 161 | 0 0 m 162 | -1 0.333 l 163 | -1 -0.333 l 164 | h 165 | -1 0 m 166 | -2 0.333 l 167 | -2 -0.333 l 168 | h 169 | 170 | 171 | 172 | 173 | 0 0 m 174 | -1 0.333 l 175 | -1 -0.333 l 176 | h 177 | -1 0 m 178 | -2 0.333 l 179 | -2 -0.333 l 180 | h 181 | 182 | 183 | 184 | 185 | 0.5 0 m 186 | -0.5 0.333 l 187 | -0.5 -0.333 l 188 | h 189 | 190 | 191 | 192 | 193 | 0.5 0 m 194 | -0.5 0.333 l 195 | -0.5 -0.333 l 196 | h 197 | 198 | 199 | 200 | 201 | 0.5 0 m 202 | -0.5 0.333 l 203 | -0.3 0 l 204 | -0.5 -0.333 l 205 | h 206 | 207 | 208 | 209 | 210 | 0.5 0 m 211 | -0.5 0.333 l 212 | -0.3 0 l 213 | -0.5 -0.333 l 214 | h 215 | 216 | 217 | 218 | 219 | 1 0 m 220 | 0 0.333 l 221 | 0 -0.333 l 222 | h 223 | 0 0 m 224 | -1 0.333 l 225 | -1 -0.333 l 226 | h 227 | 228 | 229 | 230 | 231 | 1 0 m 232 | 0 0.333 l 233 | 0 -0.333 l 234 | h 235 | 0 0 m 236 | -1 0.333 l 237 | -1 -0.333 l 238 | h 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 | 253 | 254 | 255 | 256 | 257 | 258 | 259 | 260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 | 270 | 271 | 272 | 273 | 274 | 275 | 276 | 277 | 278 | 279 | 280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 | 289 | 290 | 291 | 292 | 293 | 294 | 295 | 296 | 297 | 298 | 299 | 300 | 301 | 302 | 303 | 304 | 305 | 306 | 307 | 308 | 309 | 310 | 311 | 312 | 313 | 314 | 315 | 316 | 317 | 318 | 319 | 320 | 321 | 322 | 224 416 m 323 | 256 384 l 324 | 288 336 l 325 | 320 384 l 326 | 352 416 l 327 | 328 | 329 | 256 384 m 330 | 320 384 l 331 | 332 | 333 | 334 | -------------------------------------------------------------------------------- /Notes/MAT 6102 - Graph Theory/Images/BullGraph.ipe: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 0 0 m 9 | -1 0.333 l 10 | -1 -0.333 l 11 | h 12 | 13 | 14 | 15 | 16 | 0 0 m 17 | -1 0.333 l 18 | -1 -0.333 l 19 | h 20 | 21 | 22 | 23 | 24 | 0 0 m 25 | -1 0.333 l 26 | -0.8 0 l 27 | -1 -0.333 l 28 | h 29 | 30 | 31 | 32 | 33 | 0 0 m 34 | -1 0.333 l 35 | -0.8 0 l 36 | -1 -0.333 l 37 | h 38 | 39 | 40 | 41 | 42 | 0.6 0 0 0.6 0 0 e 43 | 0.4 0 0 0.4 0 0 e 44 | 45 | 46 | 47 | 48 | 0.6 0 0 0.6 0 0 e 49 | 50 | 51 | 52 | 53 | 54 | 0.5 0 0 0.5 0 0 e 55 | 56 | 57 | 0.6 0 0 0.6 0 0 e 58 | 0.4 0 0 0.4 0 0 e 59 | 60 | 61 | 62 | 63 | 64 | -0.6 -0.6 m 65 | 0.6 -0.6 l 66 | 0.6 0.6 l 67 | -0.6 0.6 l 68 | h 69 | -0.4 -0.4 m 70 | 0.4 -0.4 l 71 | 0.4 0.4 l 72 | -0.4 0.4 l 73 | h 74 | 75 | 76 | 77 | 78 | -0.6 -0.6 m 79 | 0.6 -0.6 l 80 | 0.6 0.6 l 81 | -0.6 0.6 l 82 | h 83 | 84 | 85 | 86 | 87 | 88 | -0.5 -0.5 m 89 | 0.5 -0.5 l 90 | 0.5 0.5 l 91 | -0.5 0.5 l 92 | h 93 | 94 | 95 | -0.6 -0.6 m 96 | 0.6 -0.6 l 97 | 0.6 0.6 l 98 | -0.6 0.6 l 99 | h 100 | -0.4 -0.4 m 101 | 0.4 -0.4 l 102 | 0.4 0.4 l 103 | -0.4 0.4 l 104 | h 105 | 106 | 107 | 108 | 109 | 110 | 111 | -0.43 -0.57 m 112 | 0.57 0.43 l 113 | 0.43 0.57 l 114 | -0.57 -0.43 l 115 | h 116 | 117 | 118 | -0.43 0.57 m 119 | 0.57 -0.43 l 120 | 0.43 -0.57 l 121 | -0.57 0.43 l 122 | h 123 | 124 | 125 | 126 | 127 | 128 | 0 0 m 129 | -1 0.333 l 130 | -1 -0.333 l 131 | h 132 | 133 | 134 | 135 | 136 | 0 0 m 137 | -1 0.333 l 138 | -0.8 0 l 139 | -1 -0.333 l 140 | h 141 | 142 | 143 | 144 | 145 | 0 0 m 146 | -1 0.333 l 147 | -0.8 0 l 148 | -1 -0.333 l 149 | h 150 | 151 | 152 | 153 | 154 | -1 0.333 m 155 | 0 0 l 156 | -1 -0.333 l 157 | 158 | 159 | 160 | 161 | 0 0 m 162 | -1 0.333 l 163 | -1 -0.333 l 164 | h 165 | -1 0 m 166 | -2 0.333 l 167 | -2 -0.333 l 168 | h 169 | 170 | 171 | 172 | 173 | 0 0 m 174 | -1 0.333 l 175 | -1 -0.333 l 176 | h 177 | -1 0 m 178 | -2 0.333 l 179 | -2 -0.333 l 180 | h 181 | 182 | 183 | 184 | 185 | 0.5 0 m 186 | -0.5 0.333 l 187 | -0.5 -0.333 l 188 | h 189 | 190 | 191 | 192 | 193 | 0.5 0 m 194 | -0.5 0.333 l 195 | -0.5 -0.333 l 196 | h 197 | 198 | 199 | 200 | 201 | 0.5 0 m 202 | -0.5 0.333 l 203 | -0.3 0 l 204 | -0.5 -0.333 l 205 | h 206 | 207 | 208 | 209 | 210 | 0.5 0 m 211 | -0.5 0.333 l 212 | -0.3 0 l 213 | -0.5 -0.333 l 214 | h 215 | 216 | 217 | 218 | 219 | 1 0 m 220 | 0 0.333 l 221 | 0 -0.333 l 222 | h 223 | 0 0 m 224 | -1 0.333 l 225 | -1 -0.333 l 226 | h 227 | 228 | 229 | 230 | 231 | 1 0 m 232 | 0 0.333 l 233 | 0 -0.333 l 234 | h 235 | 0 0 m 236 | -1 0.333 l 237 | -1 -0.333 l 238 | h 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 | 253 | 254 | 255 | 256 | 257 | 258 | 259 | 260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 | 270 | 271 | 272 | 273 | 274 | 275 | 276 | 277 | 278 | 279 | 280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 | 289 | 290 | 291 | 292 | 293 | 294 | 295 | 296 | 297 | 298 | 299 | 300 | 301 | 302 | 303 | 304 | 305 | 306 | 307 | 308 | 309 | 310 | 311 | 312 | 313 | 314 | 315 | 316 | 317 | 318 | 319 | 320 | 321 | 322 | 224 416 m 323 | 256 384 l 324 | 288 336 l 325 | 320 384 l 326 | 352 416 l 327 | 328 | 329 | 256 384 m 330 | 320 384 l 331 | 332 | 333 | 334 | -------------------------------------------------------------------------------- /Notes/MAT 2138 - Discrete Mathematics/Images/BullGraph.ipe: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 0 0 m 9 | -1 0.333 l 10 | -1 -0.333 l 11 | h 12 | 13 | 14 | 15 | 16 | 0 0 m 17 | -1 0.333 l 18 | -1 -0.333 l 19 | h 20 | 21 | 22 | 23 | 24 | 0 0 m 25 | -1 0.333 l 26 | -0.8 0 l 27 | -1 -0.333 l 28 | h 29 | 30 | 31 | 32 | 33 | 0 0 m 34 | -1 0.333 l 35 | -0.8 0 l 36 | -1 -0.333 l 37 | h 38 | 39 | 40 | 41 | 42 | 0.6 0 0 0.6 0 0 e 43 | 0.4 0 0 0.4 0 0 e 44 | 45 | 46 | 47 | 48 | 0.6 0 0 0.6 0 0 e 49 | 50 | 51 | 52 | 53 | 54 | 0.5 0 0 0.5 0 0 e 55 | 56 | 57 | 0.6 0 0 0.6 0 0 e 58 | 0.4 0 0 0.4 0 0 e 59 | 60 | 61 | 62 | 63 | 64 | -0.6 -0.6 m 65 | 0.6 -0.6 l 66 | 0.6 0.6 l 67 | -0.6 0.6 l 68 | h 69 | -0.4 -0.4 m 70 | 0.4 -0.4 l 71 | 0.4 0.4 l 72 | -0.4 0.4 l 73 | h 74 | 75 | 76 | 77 | 78 | -0.6 -0.6 m 79 | 0.6 -0.6 l 80 | 0.6 0.6 l 81 | -0.6 0.6 l 82 | h 83 | 84 | 85 | 86 | 87 | 88 | -0.5 -0.5 m 89 | 0.5 -0.5 l 90 | 0.5 0.5 l 91 | -0.5 0.5 l 92 | h 93 | 94 | 95 | -0.6 -0.6 m 96 | 0.6 -0.6 l 97 | 0.6 0.6 l 98 | -0.6 0.6 l 99 | h 100 | -0.4 -0.4 m 101 | 0.4 -0.4 l 102 | 0.4 0.4 l 103 | -0.4 0.4 l 104 | h 105 | 106 | 107 | 108 | 109 | 110 | 111 | -0.43 -0.57 m 112 | 0.57 0.43 l 113 | 0.43 0.57 l 114 | -0.57 -0.43 l 115 | h 116 | 117 | 118 | -0.43 0.57 m 119 | 0.57 -0.43 l 120 | 0.43 -0.57 l 121 | -0.57 0.43 l 122 | h 123 | 124 | 125 | 126 | 127 | 128 | 0 0 m 129 | -1 0.333 l 130 | -1 -0.333 l 131 | h 132 | 133 | 134 | 135 | 136 | 0 0 m 137 | -1 0.333 l 138 | -0.8 0 l 139 | -1 -0.333 l 140 | h 141 | 142 | 143 | 144 | 145 | 0 0 m 146 | -1 0.333 l 147 | -0.8 0 l 148 | -1 -0.333 l 149 | h 150 | 151 | 152 | 153 | 154 | -1 0.333 m 155 | 0 0 l 156 | -1 -0.333 l 157 | 158 | 159 | 160 | 161 | 0 0 m 162 | -1 0.333 l 163 | -1 -0.333 l 164 | h 165 | -1 0 m 166 | -2 0.333 l 167 | -2 -0.333 l 168 | h 169 | 170 | 171 | 172 | 173 | 0 0 m 174 | -1 0.333 l 175 | -1 -0.333 l 176 | h 177 | -1 0 m 178 | -2 0.333 l 179 | -2 -0.333 l 180 | h 181 | 182 | 183 | 184 | 185 | 0.5 0 m 186 | -0.5 0.333 l 187 | -0.5 -0.333 l 188 | h 189 | 190 | 191 | 192 | 193 | 0.5 0 m 194 | -0.5 0.333 l 195 | -0.5 -0.333 l 196 | h 197 | 198 | 199 | 200 | 201 | 0.5 0 m 202 | -0.5 0.333 l 203 | -0.3 0 l 204 | -0.5 -0.333 l 205 | h 206 | 207 | 208 | 209 | 210 | 0.5 0 m 211 | -0.5 0.333 l 212 | -0.3 0 l 213 | -0.5 -0.333 l 214 | h 215 | 216 | 217 | 218 | 219 | 1 0 m 220 | 0 0.333 l 221 | 0 -0.333 l 222 | h 223 | 0 0 m 224 | -1 0.333 l 225 | -1 -0.333 l 226 | h 227 | 228 | 229 | 230 | 231 | 1 0 m 232 | 0 0.333 l 233 | 0 -0.333 l 234 | h 235 | 0 0 m 236 | -1 0.333 l 237 | -1 -0.333 l 238 | h 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 | 253 | 254 | 255 | 256 | 257 | 258 | 259 | 260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 | 270 | 271 | 272 | 273 | 274 | 275 | 276 | 277 | 278 | 279 | 280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 | 289 | 290 | 291 | 292 | 293 | 294 | 295 | 296 | 297 | 298 | 299 | 300 | 301 | 302 | 303 | 304 | 305 | 306 | 307 | 308 | 309 | 310 | 311 | 312 | 313 | 314 | 315 | 316 | 317 | 318 | 319 | 320 | 321 | 322 | 224 416 m 323 | 256 384 l 324 | 288 336 l 325 | 320 384 l 326 | 352 416 l 327 | 328 | 329 | 256 384 m 330 | 320 384 l 331 | 332 | 333 | 334 | -------------------------------------------------------------------------------- /Notes/Applied Graph Theory/Images/Subgraphs-H1.ipe: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 0 0 m 9 | -1 0.333 l 10 | -1 -0.333 l 11 | h 12 | 13 | 14 | 15 | 16 | 0 0 m 17 | -1 0.333 l 18 | -1 -0.333 l 19 | h 20 | 21 | 22 | 23 | 24 | 0 0 m 25 | -1 0.333 l 26 | -0.8 0 l 27 | -1 -0.333 l 28 | h 29 | 30 | 31 | 32 | 33 | 0 0 m 34 | -1 0.333 l 35 | -0.8 0 l 36 | -1 -0.333 l 37 | h 38 | 39 | 40 | 41 | 42 | 0.6 0 0 0.6 0 0 e 43 | 0.4 0 0 0.4 0 0 e 44 | 45 | 46 | 47 | 48 | 0.6 0 0 0.6 0 0 e 49 | 50 | 51 | 52 | 53 | 54 | 0.5 0 0 0.5 0 0 e 55 | 56 | 57 | 0.6 0 0 0.6 0 0 e 58 | 0.4 0 0 0.4 0 0 e 59 | 60 | 61 | 62 | 63 | 64 | -0.6 -0.6 m 65 | 0.6 -0.6 l 66 | 0.6 0.6 l 67 | -0.6 0.6 l 68 | h 69 | -0.4 -0.4 m 70 | 0.4 -0.4 l 71 | 0.4 0.4 l 72 | -0.4 0.4 l 73 | h 74 | 75 | 76 | 77 | 78 | -0.6 -0.6 m 79 | 0.6 -0.6 l 80 | 0.6 0.6 l 81 | -0.6 0.6 l 82 | h 83 | 84 | 85 | 86 | 87 | 88 | -0.5 -0.5 m 89 | 0.5 -0.5 l 90 | 0.5 0.5 l 91 | -0.5 0.5 l 92 | h 93 | 94 | 95 | -0.6 -0.6 m 96 | 0.6 -0.6 l 97 | 0.6 0.6 l 98 | -0.6 0.6 l 99 | h 100 | -0.4 -0.4 m 101 | 0.4 -0.4 l 102 | 0.4 0.4 l 103 | -0.4 0.4 l 104 | h 105 | 106 | 107 | 108 | 109 | 110 | 111 | -0.43 -0.57 m 112 | 0.57 0.43 l 113 | 0.43 0.57 l 114 | -0.57 -0.43 l 115 | h 116 | 117 | 118 | -0.43 0.57 m 119 | 0.57 -0.43 l 120 | 0.43 -0.57 l 121 | -0.57 0.43 l 122 | h 123 | 124 | 125 | 126 | 127 | 128 | 0 0 m 129 | -1 0.333 l 130 | -1 -0.333 l 131 | h 132 | 133 | 134 | 135 | 136 | 0 0 m 137 | -1 0.333 l 138 | -0.8 0 l 139 | -1 -0.333 l 140 | h 141 | 142 | 143 | 144 | 145 | 0 0 m 146 | -1 0.333 l 147 | -0.8 0 l 148 | -1 -0.333 l 149 | h 150 | 151 | 152 | 153 | 154 | -1 0.333 m 155 | 0 0 l 156 | -1 -0.333 l 157 | 158 | 159 | 160 | 161 | 0 0 m 162 | -1 0.333 l 163 | -1 -0.333 l 164 | h 165 | -1 0 m 166 | -2 0.333 l 167 | -2 -0.333 l 168 | h 169 | 170 | 171 | 172 | 173 | 0 0 m 174 | -1 0.333 l 175 | -1 -0.333 l 176 | h 177 | -1 0 m 178 | -2 0.333 l 179 | -2 -0.333 l 180 | h 181 | 182 | 183 | 184 | 185 | 0.5 0 m 186 | -0.5 0.333 l 187 | -0.5 -0.333 l 188 | h 189 | 190 | 191 | 192 | 193 | 0.5 0 m 194 | -0.5 0.333 l 195 | -0.5 -0.333 l 196 | h 197 | 198 | 199 | 200 | 201 | 0.5 0 m 202 | -0.5 0.333 l 203 | -0.3 0 l 204 | -0.5 -0.333 l 205 | h 206 | 207 | 208 | 209 | 210 | 0.5 0 m 211 | -0.5 0.333 l 212 | -0.3 0 l 213 | -0.5 -0.333 l 214 | h 215 | 216 | 217 | 218 | 219 | 1 0 m 220 | 0 0.333 l 221 | 0 -0.333 l 222 | h 223 | 0 0 m 224 | -1 0.333 l 225 | -1 -0.333 l 226 | h 227 | 228 | 229 | 230 | 231 | 1 0 m 232 | 0 0.333 l 233 | 0 -0.333 l 234 | h 235 | 0 0 m 236 | -1 0.333 l 237 | -1 -0.333 l 238 | h 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 | 253 | 254 | 255 | 256 | 257 | 258 | 259 | 260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 | 270 | 271 | 272 | 273 | 274 | 275 | 276 | 277 | 278 | 279 | 280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 | 289 | 290 | 291 | 292 | 293 | 294 | 295 | 296 | 297 | 298 | 299 | 300 | 301 | 302 | 303 | 304 | 305 | 306 | 307 | 308 | 309 | 310 | 311 | 312 | 313 | 314 | 315 | 316 | 317 | 318 | 319 | 320 | v_1 321 | v_4 322 | v_7 323 | 324 | 192 416 m 325 | 288 448 l 326 | 327 | 328 | 224 448 m 329 | 192 448 l 330 | 331 | 332 | v_5 333 | v_2 334 | 335 | 336 | -------------------------------------------------------------------------------- /Notes/MAT 6102 - Graph Theory/Images/Subgraphs-H1.ipe: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 0 0 m 9 | -1 0.333 l 10 | -1 -0.333 l 11 | h 12 | 13 | 14 | 15 | 16 | 0 0 m 17 | -1 0.333 l 18 | -1 -0.333 l 19 | h 20 | 21 | 22 | 23 | 24 | 0 0 m 25 | -1 0.333 l 26 | -0.8 0 l 27 | -1 -0.333 l 28 | h 29 | 30 | 31 | 32 | 33 | 0 0 m 34 | -1 0.333 l 35 | -0.8 0 l 36 | -1 -0.333 l 37 | h 38 | 39 | 40 | 41 | 42 | 0.6 0 0 0.6 0 0 e 43 | 0.4 0 0 0.4 0 0 e 44 | 45 | 46 | 47 | 48 | 0.6 0 0 0.6 0 0 e 49 | 50 | 51 | 52 | 53 | 54 | 0.5 0 0 0.5 0 0 e 55 | 56 | 57 | 0.6 0 0 0.6 0 0 e 58 | 0.4 0 0 0.4 0 0 e 59 | 60 | 61 | 62 | 63 | 64 | -0.6 -0.6 m 65 | 0.6 -0.6 l 66 | 0.6 0.6 l 67 | -0.6 0.6 l 68 | h 69 | -0.4 -0.4 m 70 | 0.4 -0.4 l 71 | 0.4 0.4 l 72 | -0.4 0.4 l 73 | h 74 | 75 | 76 | 77 | 78 | -0.6 -0.6 m 79 | 0.6 -0.6 l 80 | 0.6 0.6 l 81 | -0.6 0.6 l 82 | h 83 | 84 | 85 | 86 | 87 | 88 | -0.5 -0.5 m 89 | 0.5 -0.5 l 90 | 0.5 0.5 l 91 | -0.5 0.5 l 92 | h 93 | 94 | 95 | -0.6 -0.6 m 96 | 0.6 -0.6 l 97 | 0.6 0.6 l 98 | -0.6 0.6 l 99 | h 100 | -0.4 -0.4 m 101 | 0.4 -0.4 l 102 | 0.4 0.4 l 103 | -0.4 0.4 l 104 | h 105 | 106 | 107 | 108 | 109 | 110 | 111 | -0.43 -0.57 m 112 | 0.57 0.43 l 113 | 0.43 0.57 l 114 | 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h 206 | 207 | 208 | 209 | 210 | 0.5 0 m 211 | -0.5 0.333 l 212 | -0.3 0 l 213 | -0.5 -0.333 l 214 | h 215 | 216 | 217 | 218 | 219 | 1 0 m 220 | 0 0.333 l 221 | 0 -0.333 l 222 | h 223 | 0 0 m 224 | -1 0.333 l 225 | -1 -0.333 l 226 | h 227 | 228 | 229 | 230 | 231 | 1 0 m 232 | 0 0.333 l 233 | 0 -0.333 l 234 | h 235 | 0 0 m 236 | -1 0.333 l 237 | -1 -0.333 l 238 | h 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 | 253 | 254 | 255 | 256 | 257 | 258 | 259 | 260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 | 270 | 271 | 272 | 273 | 274 | 275 | 276 | 277 | 278 | 279 | 280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 | 289 | 290 | 291 | 292 | 293 | 294 | 295 | 296 | 297 | 298 | 299 | 300 | 301 | 302 | 303 | 304 | 305 | 306 | 307 | 308 | 309 | 310 | 311 | 312 | 313 | 314 | 315 | 316 | 317 | 318 | 319 | 320 | v_1 321 | v_4 322 | v_7 323 | 324 | 192 416 m 325 | 288 448 l 326 | 327 | 328 | 224 448 m 329 | 192 448 l 330 | 331 | 332 | v_5 333 | v_2 334 | 335 | 336 | --------------------------------------------------------------------------------