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Lecture 1

MATH 323 Lecture Notes - Lecture 1: Memorylessness, Function Composition, Independent And Identically Distributed Random Variables


Department
Mathematics & Statistics (Sci)
Course Code
MATH 323
Professor
W.J.Anderson
Lecture
1

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INTRODUCTION
TO
PROBABILITY
William J. Anderson
McGill University

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Contents
1 Introduction and Definitions. 5
1.1 BasicDenitions..................................... 5
1.2 Permutations and Combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Conditional Probability and Independence. . . . . . . . . . . . . . . . . . . . 15
1.4 Bayes’ Rule and the Law of Total Probability. . . . . . . . . . . . . . . . . . . 20
2 Discrete Random Variables. 23
2.1 BasicDenitions..................................... 23
2.2 Special Discrete Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 The Binomial Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 The Geometric Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.3 The Negative Binomial Distribution. . . . . . . . . . . . . . . . . . . . . 30
2.2.4 The Hypergeometric Distribution. . . . . . . . . . . . . . . . . . . . . . 31
2.2.5 The Poisson Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Moment Generating Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Continuous Random Variables. 35
3.1 DistributionFunctions................................. 35
3.2 Continuous Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Special Continuous Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1 The Uniform Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.2 The Exponential Distribution. . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.3 The Gamma Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.4 The Normal Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.5 The Beta Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.6 The Cauchy Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 ChebychevsInequality................................. 50
4 Multivariate Distributions. 51
4.1 Denitions. ....................................... 51
4.2 Marginal Distributions and the Expected Value of Functions of Random
Variables. ........................................ 53
4.2.1 SpecialTheorems................................ 54
4.2.2 Covariance. ................................... 55
4.3 Conditional Probability and Density Functions. . . . . . . . . . . . . . . . . . 59
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