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of relations =2mn7. WebCPS102 DISCRETE MATHEMATICS Practice Final Exam In contrast to the homework, no collaborations are allowed. Discrete Mathematics \newcommand{\gt}{>} \newcommand{\vr}[1]{\vtx{right}{#1}} Extended form of Bayes' rule Let $\{A_i, i\in[\![1,n]\! 17 0 obj From 1 to 100, there are $50/2 = 25$ numbers which are multiples of 2. Pigeonhole Principle states that if there are fewer pigeon holes than total number of pigeons and each pigeon is put in a pigeon hole, then there must be at least one pigeon hole with more than one pigeon. Now we want to count large collections of things quickly and precisely. Number of ways of arranging the consonants among themselves $= ^3P_{3} = 3! Cumulative distribution function (CDF) The cumulative distribution function $F$, which is monotonically non-decreasing and is such that $\underset{x\rightarrow-\infty}{\textrm{lim}}F(x)=0$ and $\underset{x\rightarrow+\infty}{\textrm{lim}}F(x)=1$, is defined as: Remark: we have $P(a < X\leqslant B)=F(b)-F(a)$. Web2362 Education Cheat Sheets. (1!)(1!)(2!)] Counting problems may be hard, and easy solutions are not obvious Approach: simplify the solution by decomposing the problem Two basic decomposition rules: Product rule A count decomposes into a sequence of dependent counts (each element in the first count is associated with all elements of the second count) Sum rule Pascal's identity, first derived by Blaise Pascal in 17th century, states that the number of ways to choose k elements from n elements is equal to the summation of number of ways to choose (k-1) elements from (n-1) elements and the number of ways to choose elements from n-1 elements. If we consider two tasks A and B which are disjoint (i.e. Cheatsheet - Summary Discrete Mathematics I xmT;s1Wli+,[-:^Q1GL$E=>]KC}{~=ogwh=9-} }pNY@z }>c? of edges required = {(n-1)*(n-2)/2 } + 18. WebProof : Assume that n is an odd integer. In this case the sign means that a divides b, or that b a is an integer. 1 This is a matter of taste. In general, use the form Above Venn Diagram shows that A is a subset of B. Set DifferenceDifference between sets is denoted by A B, is the set containing elements of set A but not in B. i.e all elements of A except the element of B.ComplementThe complement of a set A, denoted by , is the set of all the elements except A. Complement of the set A is U A. GroupA non-empty set G, (G, *) is called a group if it follows the following axiom: |A| = m and |B| = n, then1. /MediaBox [0 0 612 792] A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. Solution As we are taking 6 cards at a time from a deck of 6 cards, the permutation will be $^6P_{6} = 6! Simple is harder to achieve. 3 0 obj << Here it means the absolute value of x, ie. Hence, a+c b+d(modm)andac bd(modm). Equivalesistheonlyequivalencerelationthatisassociative ((p q) r) (p (q *"TMakf9(XiBFPhr50)_9VrX3Gx"A D! He may go X to Y by either 3 bus routes or 2 train routes. From a set S ={x, y, z} by taking two at a time, all permutations are , We have to form a permutation of three digit numbers from a set of numbers $S = \lbrace 1, 2, 3 \rbrace$. Prove that if xy is irrational, then y is irrational. The number of ways to choose 3 men from 6 men is $^6C_{3}$ and the number of ways to choose 2 women from 5 women is $^5C_{2}$, Hence, the total number of ways is $^6C_{3} \times ^5C_{2} = 20 \times 10 = 200$. of ways to fill up from first place up to r-th-place , $n_{ P_{ r } } = n (n-1) (n-2).. (n-r + 1)$, $= [n(n-1)(n-2) (n-r + 1)] [(n-r)(n-r-1) \dots 3.2.1] / [(n-r)(n-r-1) \dots 3.2.1]$. A relation is an equivalence if, 1. DISCRETE MATHEMATICS FOR COMPUTER SCIENCE ~C'ZOdA3,3FHaD%B,e@,*/x}9Scv\`{]SL*|)B(u9V|My\4 Xm$qg3~Fq&M?D'Clk +&$.U;n8FHCfQd!gzMv94NU'M`cU6{@zxG,,?F,}I+52XbQN0.''f>:Vn(g."]^{\p5,`"zI%nO. of Anti Symmetric Relations = 2n*3n(n-1)/210. Education Cheat Sheets Define the set Ento be the set of binary strings with n bits that have an even number of 1's. 445 Cheatsheet - Princeton University Get up and running with ChatGPT with this comprehensive cheat sheet. To prove A is the subset of B, we need to simply show that if x belongs to A then x also belongs to B.To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B. Tree, 10. Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects. (nr+1)! set of the common element in A and B. DisjointTwo sets are said to be disjoint if their intersection is the empty set .i.e sets have no common elements. Did you make this project? WebReference Sheet for Discrete Maths PropositionalCalculus Orderofdecreasingbindingpower: =,:,^/_,)/(, /6 . I hate discrete math because its hard for me to understand. \YfM3V\d2)s/d*{C_[aaMD */N_RZ0ze2DTgCY. x3T0 BCKs=S\.t;!THcYYX endstream of irreflexive relations = 2n(n-1), 15. How many ways can you choose 3 distinct groups of 3 students from total 9 students? Therefore,b+d= (a+sm) + (c+tm) = (a+c) +m(s+t), andbd= (a+sm)(c+tm) =ac+m(at+cs+stm). /N 100 \newcommand{\st}{:} \newcommand{\B}{\mathbf B} n Less theory, more problem solving, focuses on exam problems, use as study sheet! Then m 3n 6. endobj \newcommand{\Imp}{\Rightarrow} Counting rules Discrete probability distributions In probability, a discrete distribution has either a finite or a countably infinite number of possible values. Discrete Mathematics Cheat Sheet - DocDroid Axiom 1 Every probability is between 0 and 1 included, i.e: Axiom 2 The probability that at least one of the elementary events in the entire sample space will occur is 1, i.e: Axiom 3 For any sequence of mutually exclusive events $E_1, , E_n$, we have: Permutation A permutation is an arrangement of $r$ objects from a pool of $n$ objects, in a given order. \). /ImageMask true If there are n elements of which $a_1$ are alike of some kind, $a_2$ are alike of another kind; $a_3$ are alike of third kind and so on and $a_r$ are of $r^{th}$ kind, where $(a_1 + a_2 + a_r) = n$. 6 0 obj I'll check out your sheet when I get to my computer. To guarantee that a graph with n vertices is connected, minimum no. /Width 156 endobj Here, the ordering does not matter. (c) Express P(k + 1). 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 Combinatorial Proofs 1.5 Stars and Bars 1.6 Advanced Counting Using PIE { (k-1)!(n-k)! } NOTE: Order of elements of a set doesnt matter. Let G be a connected planar simple graph with n vertices and m edges, and no triangles. /Filter /FlateDecode WebE(X)=xP(X=x) (for discreteX) x 1 E(X) =xf(x)dx(for continuousX) TheLaw of the Unconscious Statistician (LOTUS)states thatyou can nd the expected value of afunction of a random It wasn't meant to be a presentation per se, but more of a study sheet, so I did not work too hard on the typesetting. From there, he can either choose 4 bus routes or 5 train routes to reach Z. 1 Sets and Lists 2 Binomial Coefcients 3 Equivalence Relations Homework Assignments 4 1 Sets and Lists \newcommand{\N}{\mathbb N} 1.1 Additive and Multiplicative Principles 1.2 Binomial Coefficients 1.3 Combinations and Permutations 1.4 xS@}WD"f<7.\$.iH(Rc'vbo*g1@9@I4_ F2 }3^C2>2B@>8JfWkn%;?t!yb C;.AIyir!zZn}Na;$t"2b {HEx}]Zg;'B!e>3B=DWw,qS9\ THi_WI04$-1cb Sum of degree of all vertices is equal to twice the number of edges.4. endobj [/Pattern /DeviceRGB] It is determined as follows: Characteristic function A characteristic function $\psi(\omega)$ is derived from a probability density function $f(x)$ and is defined as: Euler's formula For $\theta \in \mathbb{R}$, the Euler formula is the name given to the identity: Revisiting the $k^{th}$ moment The $k^{th}$ moment can also be computed with the characteristic function as follows: Transformation of random variables Let the variables $X$ and $Y$ be linked by some function. Binomial Coecients 75 5.5. <> By noting $f_X$ and $f_Y$ the distribution function of $X$ and $Y$ respectively, we have: Leibniz integral rule Let $g$ be a function of $x$ and potentially $c$, and $a, b$ boundaries that may depend on $c$. WebIn the following sections, we are going to keep the same notations as before and the formulas will be explicitly detailed for the discrete (D) and continuous (C) cases. $|A \cup B| = |A| + |B| - |A \cap B| = 25 + 16 - 8 = 33$. No. :oCH7ZG_ (SO/ FXe'%Dc,1@dEAeQj]~A+H~KdF'#.(5?w?EmD9jv|H ?K?*]ZrLbu7,J^(80~*@dL"rjx /ca 1.0 = 6$ ways. We have: Covariance We define the covariance of two random variables $X$ and $Y$, that we note $\sigma_{XY}^2$ or more commonly $\textrm{Cov}(X,Y)$, as follows: Correlation By noting $\sigma_X, \sigma_Y$ the standard deviations of $X$ and $Y$, we define the correlation between the random variables $X$ and $Y$, noted $\rho_{XY}$, as follows: Remark 1: we note that for any random variables $X, Y$, we have $\rho_{XY}\in[-1,1]$. WebStep 1: Discrete Math Cram Sheet/Cheat Sheet/Study Sheet/Study Guide in PDF. @ys(5u$E$VY(@[Y+J(or(0ze7+s([nlY+J(or(0zemFGn2+%f mEH(X Part1.Indicatewhethertheargumentisvalidorinvalid.Forvalid arguments,provethattheargumentisvalidusingatruthtable.For invalid arguments, give truth values for the variables showing that the argument is. Download the PDF version here. \newcommand{\Iff}{\Leftrightarrow} Discrete Math Cheat Sheet by Dois - Cheatography Cheat Sheet of Mathemtical Notation and Terminology Examples:x:= 5means thatxis dened to be5, orf.x/ :=x2 *1means that the functionf is dened to bex2 * 1, orA:= ^1;5;7means that the setAis dened to The Pigeonhole Principle 77 Chapter 6. I have a class in it right now actually! of one to one function = (n, P, m)3. I dont know whether I agree with the name, but its a nice cheat sheet. We say that $\{A_i\}$ is a partition if we have: Remark: for any event $B$ in the sample space, we have $\displaystyle P(B)=\sum_{i=1}^nP(B|A_i)P(A_i)$. By using this website, you agree with our Cookies Policy. 14 0 obj xm=j0 gRR*9BGRGF. \newcommand{\Q}{\mathbb Q} Ten men are in a room and they are taking part in handshakes. WebDiscrete Math Review n What you should know about discrete math before the midterm. Bnis the set of binary strings with n bits. \newcommand{\imp}{\rightarrow} Discrete Mathematics Cheat Sheet I strongly believe that simple is better than complex. /AIS false In daily lives, many a times one needs to find out the number of all possible outcomes for a series of events. WebThe ultimate cheat sheet - the shortest possible document which basically covers all of maths from say algebra to whatever comes after calculus. this looks promising :), Reply For complete graph the no . 23 0 obj << { k!(n-k-1)! Discrete Math Cheat Sheet by Dois via cheatography.com/11428/cs/1340/ Complex Numbers j = -1 j = -j j = 1 z = a + bj z = r(sin + jsin) z = re tan b/a = A cos a/r Power SetsThe power set is the set all possible subset of the set S. Denoted by P(S).Example: What is the power set of {0, 1, 2}?Solution: All possible subsets{}, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}.Note: Empty set and set itself is also the member of this set of subsets. \newcommand{\vl}[1]{\vtx{left}{#1}} <> Solution There are 3 vowels and 3 consonants in the word 'ORANGE'. Reference Sheet for Discrete Maths - GitHub Pages See Last Minute Notes on all subjects here. Probability 78 Chapter 7. Proof Let there be n different elements. Question A boy lives at X and wants to go to School at Z. Pascal's identity, first derived by Blaise Pascal in 17 century, states that Once we can count, we can determine the likelihood of a particular even and we can estimate how long a computer algorithm takes to complete a task. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. English to French cheat sheet, with useful words and phrases to take with you on holiday. << \renewcommand{\iff}{\leftrightarrow} Let s = q + r and s = e f be written in lowest terms. acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Data Structures & Algorithms in JavaScript, Data Structure & Algorithm-Self Paced(C++/JAVA), Full Stack Development with React & Node JS(Live), Android App Development with Kotlin(Live), Python Backend Development with Django(Live), DevOps Engineering - Planning to Production, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Mathematics | Introduction to Propositional Logic | Set 1, Discrete Mathematics Applications of Propositional Logic, Difference between Propositional Logic and Predicate Logic, Mathematics | Predicates and Quantifiers | Set 1, Mathematics | Some theorems on Nested Quantifiers, Mathematics | Set Operations (Set theory), Mathematics | Sequence, Series and Summations, Mathematics | Representations of Matrices and Graphs in Relations, Mathematics | Introduction and types of Relations, Mathematics | Closure of Relations and Equivalence Relations, Permutation and Combination Aptitude Questions and Answers, Discrete Maths | Generating Functions-Introduction and Prerequisites, Inclusion-Exclusion and its various Applications, Project Evaluation and Review Technique (PERT), Mathematics | Partial Orders and Lattices, Mathematics | Probability Distributions Set 1 (Uniform Distribution), Mathematics | Probability Distributions Set 2 (Exponential Distribution), Mathematics | Probability Distributions Set 3 (Normal Distribution), Mathematics | Probability Distributions Set 5 (Poisson Distribution), Mathematics | Graph Theory Basics Set 1, Mathematics | Walks, Trails, Paths, Cycles and Circuits in Graph, Mathematics | Independent Sets, Covering and Matching, How to find Shortest Paths from Source to all Vertices using Dijkstras Algorithm, Introduction to Tree Data Structure and Algorithm Tutorials, Prims Algorithm for Minimum Spanning Tree (MST), Kruskals Minimum Spanning Tree (MST) Algorithm, Tree Traversals (Inorder, Preorder and Postorder), Travelling Salesman Problem using Dynamic Programming, Check whether a given graph is Bipartite or not, Eulerian path and circuit for undirected graph, Fleurys Algorithm for printing Eulerian Path or Circuit, Chinese Postman or Route Inspection | Set 1 (introduction), Graph Coloring | Set 1 (Introduction and Applications), Check if a graph is Strongly, Unilaterally or Weakly connected, Handshaking Lemma and Interesting Tree Properties, Mathematics | Rings, Integral domains and Fields, Topic wise multiple choice questions in computer science, A graph is planar if and only if it does not contain a subdivision of K. Let G be a connected planar graph, and let n, m and f denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of G. Then n m + f = 2. '1g[bXlF) q^|W*BmHYGd tK5A+(R%9;P@2[P9?j28C=r[%\%U08$@`TaqlfEYCfj8Zx!`,O%L v+ ]F$Dx U. The permutation will be = 123, 132, 213, 231, 312, 321, The number of permutations of n different things taken r at a time is denoted by $n_{P_{r}}$. DISCRETE MATHEMATICS FOR COMPUTER SCIENCE - Duke xKs6. \renewcommand{\bar}{\overline} It is determined as follows: Standard deviation The standard deviation of a random variable, often noted $\sigma$, is a measure of the spread of its distribution function which is compatible with the units of the actual random variable. \[\boxed{P\left(\bigcup_{i=1}^nE_i\right)=\sum_{i=1}^nP(E_i)}\], \[\boxed{C(n, r)=\frac{P(n, r)}{r!}=\frac{n!}{r!(n-r)! discrete math counting cheat sheet.pdf - | Course Hero Graph Theory 82 7.1. % xY8_1ow>;|D@`a%e9l96=u=uQ 3 and m edges. U denotes the universal set. Then, number of permutations of these n objects is = $n! I go out of my way to simplify subjects. For example: In a group of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes took place? <> \dots (a_r!)]$. Before tackling questions like these, let's look at the basics of counting. Affordable solution to train a team and make them project ready. Once we can count, we can determine the likelihood of a particular even and we can estimate how long a No. Note that in this case it is written \mid in LaTeX, and not with the symbol |. /Filter /FlateDecode WebThe first principle of counting involves the student using a list of words to count in a repeatable order. /Height 25 >> endobj Corollary Let m be a positive integer and let a and b be integers. /Type /ObjStm on Introduction. of the domain. There are 6 men and 5 women in a room. Share it with us! on April 20, 2023, 5:30 PM EDT. WebSincea b(modm)andc d(modm), by the Theorem abovethere are integerssandt withb=a+smandd=c+tm. Discrete mathematics cheat sheet Last Minute Notes Discrete Mathematics - GeeksforGeeks xVO8~_1o't?b'jr=KhbUoEj|5%$$YE?I:%a1JH&$rA?%IjF d In a group of 50 students 24 like cold drinks and 36 like hot drinks and each student likes at least one of the two drinks. The Rule of Sum If a sequence of tasks $T_1, T_2, \dots, T_m$ can be done in $w_1, w_2, \dots w_m$ ways respectively (the condition is that no tasks can be performed simultaneously), then the number of ways to do one of these tasks is $w_1 + w_2 + \dots +w_m$. Discrete Mathematics - Counting Theory. How many different 10 lettered PAN numbers can be generated such that the first five letters are capital alphabets, the next four are digits and the last is again a capital letter. For solving these problems, mathematical theory of counting are used. Counting mainly encompasses fundamental counting rule, 592 Heres something called a theoretical computer science cheat sheet. + \frac{ (n-1)! } ]$, The number of circular permutations of n different elements taken x elements at time = $^np_{x}/x$, The number of circular permutations of n different things = $^np_{n}/n$. /SMask /None>> Rsolution chap02 - Corrig du chapitre 2 de benson Physique 2; CCNA 1 v7 Modules 16 17 Building and Securing a Small Network Exam Answers; Processing and value addition in ornamental flower crops (2019-AJ-66) Chapitre 3 r ponses (STE) Homework 9.3 FWfSE xpwy8+3o We can now generalize the number of ways to fill up r-th place as [n (r1)] = nr+1, So, the total no. Notes on Discrete Mathematics /Length 1235 (d) In an inductive proof that for every positive integer n, Let B = {0, 1}. << So, $| X \cup Y | = 50$, $|X| = 24$, $|Y| = 36$, $|X \cap Y| = |X| + |Y| - |X \cup Y| = 24 + 36 - 50 = 60 - 50 = 10$. WebBefore tackling questions like these, let's look at the basics of counting. gQVmDYm*% QKP^n,D%7DBZW=pvh#(sG $A \cap B = \emptyset$), then mathematically $|A \cup B| = |A| + |B|$, The Rule of Product If a sequence of tasks $T_1, T_2, \dots, T_m$ can be done in $w_1, w_2, \dots w_m$ ways respectively and every task arrives after the occurrence of the previous task, then there are $w_1 \times w_2 \times \dots \times w_m$ ways to perform the tasks. How many anagrams are there of anagram? /First 812 2195 /Contents 3 0 R on Introduction. c o m) Paths and Circuits 91 3 /Filter /FlateDecode `y98R uA>?2 AJ|tuuU7s:_/R~faGuC7c_lqxt1~6!Xb2{gsoLFy"TJ4{oXbECVD-&}@~O@8?ARX/M)lJ4D(7! Complemented Lattice : Every element has complement17. << Hence, there are (n-1) ways to fill up the second place. in the word 'READER'. &@(BR-c)#b~9md@;iR2N {\TTX|'Wv{KdB?Hs}n^wVWZND+->TLqzZt,[kS3#P:OJ6NzW"OR]a'Q~%>6 28 0 obj << 8"NE!OI6%pu=s{ZW"c"(E89/48q endobj \newcommand{\R}{\mathbb R} Graph Theory; Notes on Counting; Notes on Distributions and Stirling numbers of the second kind; Notes on Cardinality of Sets; Notes on the Pigeonhole Principle; Notes on Combinatorial Arguments; Notes on Recurrence Relations; Notes on Inclusion-Exclusion; Notes on Generating Functions /SA true /Type /Page SA+9)UI)bwKJGJ-4D tFX9LQ The order of elements does not matter in a combination.which gives us-, Binomial Coefficients: The -combinations from a set of elements if denoted by . stream No. /Parent 22 0 R The Inclusion-exclusion principle computes the cardinal number of the union of multiple non-disjoint sets. Cram sheet/Cheat sheet/study sheet for a discrete math class that covers sequences, recursive formulas, summation, logic, sets, power sets, functions, combinatorics, arrays and matrices. If n pigeons are put into m pigeonholes where n > m, there's a hole with more than one pigeon. o[rgQ *q$E$Y:CQJ.|epOd&\AT"y@$X It includes the enumeration or counting of objects having certain properties. Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. @>%c0xC8a%k,s;b !AID/~ /Decode [1 0] Learn more. Discrete Math Cram Sheet - Ateneo de Manila University By using our site, you It is computed as follows: Remark: the $k^{th}$ moment is a particular case of the previous definition with $g:X\mapsto X^k$. The cardinality of the set is 6 and we have to choose 3 elements from the set. Problem 2 In how many ways can the letters of the word 'READER' be arranged? For choosing 3 students for 1st group, the number of ways $^9C_{3}$, The number of ways for choosing 3 students for 2nd group after choosing 1st group $^6C_{3}$, The number of ways for choosing 3 students for 3rd group after choosing 1st and 2nd group $^3C_{3}$, Hence, the total number of ways $= ^9C_{3} \times ^6C_{3} \times ^3C_{3} = 84 \times 20 \times 1 = 1680$. Basic Principles 69 5.2. A poset is called Lattice if it is both meet and join semi-lattice16. ?,%"oa)bVFQlBb60f]'1lRY/@qtNK[InziP Yh2Ng/~1]#rcpI!xHMK)1zX.F+2isv4>_Jendstream WebBefore tackling questions like these, let's look at the basics of counting. Distributive Lattice : Every Element has zero or 1 complement .18. /Filter /FlateDecode 'A`zH9sOoH=%()+[|%+&w0L1UhqIiU\|IwVzTFGMrRH3xRH`zQAzz`l#FSGFY'PS$'IYxu^v87(|q?rJ("?u1#*vID =HA`miNDKH;8&.2_LcVfgsIVAxx$A,t([k9QR$jmOX#Q=s'0z>SUxH-5OPuVq+"a;F} For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20. Representations of Graphs 88 7.3. Hence from X to Z he can go in $5 \times 9 = 45$ ways (Rule of Product). endobj From a night class at Fordham University, NYC, Fall, 2008. \newcommand{\U}{\mathcal U} Generalized Permutations and Combinations 73 5.4. /Font << /F17 6 0 R /F18 9 0 R /F15 12 0 R /F7 15 0 R /F8 18 0 R /F37 21 0 R >> The remaining 3 vacant places will be filled up by 3 vowels in $^3P_{3} = 3! ]\}$ be a partition of the sample space. \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} Let X be the set of students who like cold drinks and Y be the set of people who like hot drinks. Probability density function (PDF) The probability density function $f$ is the probability that $X$ takes on values between two adjacent realizations of the random variable. }$, $= (n-1)! Combinatorics 71 5.3. This implies that there is some integer k such that n = 2k + 1. /Type /Page Discrete Mathematics Applications of Propositional Logic; Difference between Propositional Logic and Predicate Logic; Mathematics | Propositional &IP")0 QlaK5 )CPq 9n TVd,L j' )3 O@ 3+$ >+:>Ov?! If the outcome of the experiment is contained in $E$, then we say that $E$ has occurred. 3 0 obj For $k, \sigma>0$, we have the following inequality: Discrete distributions Here are the main discrete distributions to have in mind: Continuous distributions Here are the main continuous distributions to have in mind: Joint probability density function The joint probability density function of two random variables $X$ and $Y$, that we note $f_{XY}$, is defined as follows: Marginal density We define the marginal density for the variable $X$ as follows: Cumulative distribution We define cumulative distrubution $F_{XY}$ as follows: Conditional density The conditional density of $X$ with respect to $Y$, often noted $f_{X|Y}$, is defined as follows: Independence Two random variables $X$ and $Y$ are said to be independent if we have: Moments of joint distributions We define the moments of joint distributions of random variables $X$ and $Y$ as follows: Distribution of a sum of independent random variables Let $Y=X_1++X_n$ with $X_1, , X_n$ independent. Then, The binomial expansion using Combinatorial symbols. Cartesian ProductsLet A and B be two sets. \newcommand{\Z}{\mathbb Z} Cartesian product of A and B is denoted by A B, is the set of all ordered pairs (a, b), where a belong to A and b belong to B. \renewcommand{\v}{\vtx{above}{}} ("#} &. Thank you - hope it helps. Discrete Mathematics 1.Implication : 2.Converse : The converse of the proposition is 3.Contrapositive : The contrapositive of the proposition is 4.Inverse : The inverse of the proposition is. Mathematics | Combinatorics Basics | x |. A permutation is an arrangement of some elements in which order matters. No. How many ways are there to go from X to Z? Discrete Mathematics Let G be a connected planar simple graph with n vertices, where n ? of edges to have connected graph with n vertices = n-17. Thus, n2 is odd. ]8$_v'6\2V1A) cz^U@2"jAS?@nF'8C!g1ZF%54fI4HIs e"@hBN._4~[E%V?#heH1P|'?0D#jX4Ike+{7fmc"Y$c1Fj%OIRr2^0KS)6,u`k*2D8X~@ @49d)S!Y+ad~T3=@YA )w[Il35yNrk!3PdsoZ@iqFd39|x;MUqK.-DbV]kx7VqD[h6Y[r]sd}?%endstream No. /Resources 23 0 R So, $|A|=25$, $|B|=16$ and $|A \cap B|= 8$. It is computed as follows: Generalization of the expected value The expected value of a function of a random variable $g(X)$ is computed as follows: $k^{th}$ moment The $k^{th}$ moment, noted $E[X^k]$, is the value of $X^k$ that we expect to observe on average on infinitely many trials. \newcommand{\inv}{^{-1}} mathematics In how many ways we can choose 3 men and 2 women from the room? Probability Cheatsheet v2.0 Thinking Conditionally Law of Cram sheet/Cheat sheet/study sheet for a discrete math class that covers sequences, = 180.$. Counting Principles - Counting and Cardinality Boolean Lattice: It should be both complemented and distributive. No. %PDF-1.3 *3-d[\HxSi9KpOOHNn uiKa, Counting - Discrete Mathematics Prove the following using a proof by contrapositive: Let x be a rational number. After filling the first place (n-1) number of elements is left. \newcommand{\vb}[1]{\vtx{below}{#1}}

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