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     STAT 220/230/240 COURSE NOTES By Chris Springer. Revised by Jerry Lawless, Don McLeish and Cyntha Struthers.Fall 2016 Edition  Contents 1. Introduction to Probability 1 1.1 Definitions of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problems on Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Mathematical Probability Models 5 2.1 Sample Spaces and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Problems on Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3. Probability – Counting Techniques 15 3.1 Counting Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Problems on Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4. Probability Rules and Conditional Probability 36 4.1 General Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Rules for Unions of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Intersections of Events and Independence . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.5 Product Rules, Law of Total Probability and Bayes’ Theorem . . . . . . . . . . . . . . 534.6 Useful Series and Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.7 Problems on Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5. Discrete Random Variables and Probability Models 69 5.1 Random Variables and Probability Functions . . . . . . . . . . . . . . . . . . . . . . . 695.2 Discrete Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.3 Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.4 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.5 Negative Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.6 Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87ii  CONTENTS  iii5.7 Poisson Distribution from Binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.8 Poisson Distribution from Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . 915.9 Combining Other Models with the Poisson Process . . . . . . . . . . . . . . . . . . . 965.10 Summary of Single Variable Discrete Models . . . . . . . . . . . . . . . . . . . . . . 985.11 Problems on Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6. Computational Methods and  R  107 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3 Arithmetic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.4 Some Basic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.5 R Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.6 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.7 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.8 Problems on Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7. Expected Value and Variance 116 7.1 Summarizing Data on Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 1167.2 Expectation of a Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.3 Some Applications of Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.4 Means and Variances of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.5 Problems on Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8. Continuous Probability Distributions 142 8.1 General Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428.2 Continuous Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.3 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578.4 A Method for Computer Generation of Random Variables . . . . . . . . . . . . . . . . 1638.5 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.6 Problems on Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9. Multivariate Distributions 184 9.1 Basic Terminology and Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1849.2 Multinomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1929.3 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1999.4 Expectation for Multivariate Distributions: Covariance and Correlation . . . . . . . . . 2059.5 Mean and Variance of a Linear Combination of Random Variables . . . . . . . . . . . 214
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