A First Course in Probability

A First Course in Probability, Eighth Edition , features clear and intuitive explanations of the mathematics of probability theory, outstanding problem sets, and a variety of diverse examples and applications. This book is ideal for an upper-level undergraduate or graduate level introduction to probability for math, science, engineering and business students. It assumes a backg...

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A First Course in Probability, Eighth Edition , features clear and intuitive explanations of the mathematics of probability theory, outstanding problem sets, and a variety of diverse examples and applications. This book is ideal for an upper-level undergraduate or graduate level introduction to probability for math, science, engineering and business students. It assumes a background in elementary calculus.

Sheldon M. Ross is a professor in the Department of Industrial Engineering and Operations Research at the University of Southern California. He received his Ph.D. in statistics at Stanford University in 1968. He has published many technical articles and textbooks in the areas of statistics and applied probability. Among his texts are A First Course in Probability, Introduction ...

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Sheldon M. Ross is a professor in the Department of Industrial Engineering and Operations Research at the University of Southern California. He received his Ph.D. in statistics at Stanford University in 1968. He has published many technical articles and textbooks in the areas of statistics and applied probability. Among his texts are A First Course in Probability, Introduction to Probability Models, Stochastic Processes, and Introductory Statistics. Professor Ross is the founding and continuing editor of the journal Probability in the Engineering and Informational Sciences, the Advisory Editor for International Journal of Quality Technology and Quantitative Management, and an Editorial Board Member of the Journal of Bond Trading and Management. He is a Fellow of the Institute of Mathematical Statistics and a recipient of the Humboldt US Senior Scientist Award.

Contents
Preface xi
1 Combinatorial Analysis 1
1.1 Introduction . . . .............................. 1
1.2 The Basic Principle of Counting . . . ................... 1
1.3 Permutations................................. 3
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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
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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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