A First Course in Probability

Contents Preface xi 1 Combinatorial Analysis 1 1.1 Introduction . . . .............................. 1 1.2 The Basic Principle of Counting . . . ................... 1 1.3 Permutations................................. 3 1.4 Combinations . . .............................. 5 1.5 Multinomial Coefficients . . . ....................... 9 1.6 The Number of Integer Solutions of Equations . ............ 12 Summary . .................................. 15 Problems ................................... 16 Theoretical Exercises . . . . . ....................... 18 Self-Test Problems and Exercises . . ................... 20 2 Axioms of Probability 22 2.1 Introduction . . . .............................. 22 2.2 Sample Space and Events.......................... 22 2.3 Axioms of Probability . . . . . ....................... 26 2.4 Some Simple Propositions . . ....................... 29 2.5 Sample Spaces Having Equally Likely Outcomes ............ 33 2.6 Probability as a Continuous Set Function . . . . . ............ 44 2.7 Probability as a Measure of Belief . . ................... 48 Summary . .................................. 49 Problems ................................... 50 Theoretical Exercises . . . . . ....................... 54 Self-Test Problems and Exercises . . ................... 56 3 Conditional Probability and Independence 58 3.1 Introduction . . . .............................. 58 3.2 Conditional Probabilities . . . ....................... 58 3.3 Bayes’s Formula . .............................. 65 3.4 IndependentEvents............................. 79 3.5 P (· |F ) Is a Probability . . . . . ....................... 93 Summary . .................................. 101 Problems ................................... 102 Theoretical Exercises . . . . . ....................... 110 Self-Test Problems and Exercises . . ................... 114 4 Random Variables 117 4.1 Random Variables .............................. 117 4.2 Discrete Random Variables . ....................... 123 4.3 Expected Value ............................... 125 4.4 Expectation of a Function of a Random Variable ............ 128 4.5 Variance . .................................. 132 4.6 The Bernoulli and Binomial Random Variables . ............ 134 4.6.1 Properties of Binomial Random Variables ............ 139 4.6.2 Computing the Binomial Distribution Function . . . . ..... 142 vii viii Contents 4.7 The Poisson Random Variable ....................... 143 4.7.1 Computing the Poisson Distribution Function . . . . . ..... 154 4.8 Other Discrete Probability Distributions . . . . . ............ 155 4.8.1 The Geometric Random Variable . . . . . ............ 155 4.8.2 The Negative Binomial Random Variable ............ 157 4.8.3 The Hypergeometric Random Variable . ............ 160 4.8.4 TheZeta(orZipf)Distribution.................. 163 4.9 Expected Value of Sums of Random Variables . ............ 164 4.10 Properties of the Cumulative Distribution Function . . . . . ...... 168 Summary . .................................. 170 Problems ................................... 172 Theoretical Exercises . . . . . ....................... 179 Self-Test Problems and Exercises . . ................... 183 5 Continuous Random Variables 186 5.1 Introduction . . . .............................. 186 5.2 Expectation and Variance of Continuous Random Variables ..... 190 5.3 The Uniform Random Variable . . . ................... 194 5.4 Normal Random Variables . . ....................... 198 5.4.1 The Normal Approximation to the Binomial Distribution . . . 204 5.5 Exponential Random Variables . . . ................... 208 5.5.1 Hazard Rate Functions ....................... 212 5.6 Other Continuous Distributions . . . ................... 215 5.6.1 The Gamma Distribution ..................... 215 5.6.2 The Weibull Distribution ..................... 216 5.6.3 The Cauchy Distribution...................... 217 5.6.4 The Beta Distribution ....................... 218 5.7 The Distribution of a Function of a Random Variable . . . ...... 219 Summary . .................................. 222 Problems ................................... 224 Theoretical Exercises . . . . . ....................... 227 Self-Test Problems and Exercises . . ................... 229 6 Jointly Distributed Random Variables 232 6.1 Joint Distribution Functions ........................ 232 6.2 Independent Random Variables . . . ................... 240 6.3 Sums of Independent Random Variables . . . . . ............ 252 6.3.1 Identically Distributed Uniform Random Variables . ..... 252 6.3.2 Gamma Random Variables . ................... 254 6.3.3 Normal Random Variables . ................... 256 6.3.4 Poisson and Binomial Random Variables ............ 259 6.3.5 Geometric Random Variables ................... 260 6.4 Conditional Distributions: Discrete Case . . . . . ............ 263 6.5 Conditional Distributions: Continuous Case . . . ............ 266 6.6 Order Statistics ............................... 270 6.7 Joint Probability Distribution of Functions of Random Variables . . . 274 6.8 Exchangeable Random Variables . . ................... 282 Summary . .................................. 285 Problems ................................... 287 Theoretical Exercises . . . . . ....................... 291 Self-Test Problems and Exercises . . ................... 293 Contents ix 7 Properties of Expectation 297 7.1 Introduction . . . .............................. 297 7.2 Expectation of Sums of Random Variables . . . . ............ 298 7.2.1 Obtaining Bounds from Expectations via the Probabilistic Method .................... 311 7.2.2 The Maximum–Minimums Identity . . . . ............ 313 7.3 Moments of the Number of Events that Occur . . ............ 315 7.4 Covariance, Variance of Sums, and Correlations . ............ 322 7.5 Conditional Expectation . . . ....................... 331 7.5.1 Definitions.............................. 331 7.5.2 Computing Expectations by Conditioning ............ 333 7.5.3 Computing Probabilities by Conditioning ............ 344 7.5.4 Conditional Variance . ....................... 347 7.6 Conditional Expectation and Prediction . . . . . ............ 349 7.7 Moment Generating Functions ....................... 354 7.7.1 Joint Moment Generating Functions . . . ............ 363 7.8 Additional Properties of Normal Random Variables . . . . ...... 365 7.8.1 The Multivariate Normal Distribution . . ............ 365 7.8.2 The Joint Distribution of the Sample Mean and Sample Variance ........................ 367 7.9 General Definition of Expectation . . ................... 369 Summary . .................................. 370 Problems ................................... 373 Theoretical Exercises . . . . . ....................... 380 Self-Test Problems and Exercises . . ................... 384 8 Limit Theorems 388 8.1 Introduction . . . .............................. 388 8.2 Chebyshev’s Inequality and the Weak Law of Large Numbers . .................................. 388 8.3 TheCentralLimitTheorem ........................ 391 8.4 The Strong Law of Large Numbers . ................... 400 8.5 Other Inequalities .............................. 403 8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson Random Variable .............................. 410 Summary . .................................. 412 Problems ................................... 412 Theoretical Exercises . . . . . ....................... 414 Self-Test Problems and Exercises . . ................... 415 9 Additional Topics in Probability 417 9.1 The Poisson Process . . . . . . ....................... 417 9.2 Markov Chains................................ 419 9.3 Surprise, Uncertainty, and Entropy . ................... 425 9.4 Coding Theory and Entropy . ....................... 428 Summary . .................................. 434 Problems and Theoretical Exercises . ................... 435 Self-Test Problems and Exercises . . ................... 436 References .................................. 436 x Contents 10 Simulation 438 10.1 Introduction . . . .............................. 438 10.2 General Techniques for Simulating Continuous Random Variables . . 440 10.2.1 The Inverse Transformation Method . . . ............ 441 10.2.2 The Rejection Method ....................... 442 10.3 Simulating from Discrete Distributions . . . . . . ............ 447 10.4 Variance Reduction Techniques . . . ................... 449 10.4.1 Use of Antithetic Variables . ................... 450 10.4.2 Variance Reduction by Conditioning . . . ............ 451 10.4.3 Control Variates . . . ....................... 452 Summary . .................................. 453 Problems ................................... 453 Self-Test Problems and Exercises . . ................... 455 Reference .................................. 455 Answers to Selected Problems 457 Solutions to Self-Test Problems and Exercises 461 Index

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