A First Course in Probability, 8th Edition

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 8
Preface......Page 12
1.2 The Basic Principle of Counting......Page 16
1.3 Permutations......Page 18
1.4 Combinations......Page 20
1.5 Multinomial Coefficients......Page 24
1.6 The Number of Integer Solutions of Equations......Page 27
Summary......Page 30
Problems......Page 31
Theoretical Exercises......Page 33
Self-Test Problems and Exercises......Page 35
2.2 Sample Space and Events......Page 37
2.3 Axioms of Probability......Page 41
2.4 Some Simple Propositions......Page 44
2.5 Sample Spaces Having Equally Likely Outcomes......Page 48
2.6 Probability as a Continuous Set Function......Page 59
2.7 Probability as a Measure of Belief......Page 63
Summary......Page 64
Problems......Page 65
Theoretical Exercises......Page 69
Self-Test Problems and Exercises......Page 71
3.2 Conditional Probabilities......Page 73
3.3 Bayes's Formula......Page 80
3.4 Independent Events......Page 94
3.5 P(·|F) Is a Probability......Page 108
Summary......Page 116
Problems......Page 117
Theoretical Exercises......Page 125
Self-Test Problems and Exercises......Page 129
4.1 Random Variables......Page 132
4.2 Discrete Random Variables......Page 138
4.3 Expected Value......Page 140
4.4 Expectation of a Function of a Random Variable......Page 143
4.5 Variance......Page 147
4.6 The Bernoulli and Binomial Random Variables......Page 149
4.6.1 Properties of Binomial Random Variables......Page 154
4.6.2 Computing the Binomial Distribution Function......Page 157
4.7 The Poisson Random Variable......Page 158
4.7.1 Computing the Poisson Distribution Function......Page 169
4.8.1 The Geometric Random Variable......Page 170
4.8.2 The Negative Binomial Random Variable......Page 172
4.8.3 The Hypergeometric Random Variable......Page 175
4.8.4 The Zeta (or Zipf) Distribution......Page 178
4.9 Expected Value of Sums of Random Variables......Page 179
4.10 Properties of the Cumulative Distribution Function......Page 183
Summary......Page 185
Problems......Page 187
Theoretical Exercises......Page 194
Self-Test Problems and Exercises......Page 198
5.1 Introduction......Page 201
5.2 Expectation and Variance of Continuous Random Variables......Page 205
5.3 The Uniform Random Variable......Page 209
5.4 Normal Random Variables......Page 213
5.4.1 The Normal Approximation to the Binomial Distribution......Page 219
5.5 Exponential Random Variables......Page 223
5.5.1 Hazard Rate Functions......Page 227
5.6.1 The Gamma Distribution......Page 230
5.6.2 The Weibull Distribution......Page 231
5.6.3 The Cauchy Distribution......Page 232
5.6.4 The Beta Distribution......Page 233
5.7 The Distribution of a Function of a Random Variable......Page 234
Summary......Page 237
Problems......Page 239
Theoretical Exercises......Page 242
Self-Test Problems and Exercises......Page 244
6.1 Joint Distribution Functions......Page 247
6.2 Independent Random Variables......Page 255
6.3.1 Identically Distributed Uniform Random Variables......Page 267
6.3.2 Gamma Random Variables......Page 269
6.3.3 Normal Random Variables......Page 271
6.3.4 Poisson and Binomial Random Variables......Page 274
6.3.5 Geometric Random Variables......Page 275
6.4 Conditional Distributions: Discrete Case......Page 278
6.5 Conditional Distributions: Continuous Case......Page 281
6.6 Order Statistics......Page 285
6.7 Joint Probability Distribution of Functions of Random Variables......Page 289
6.8 Exchangeable Random Variables......Page 297
Summary......Page 300
Problems......Page 302
Theoretical Exercises......Page 306
Self-Test Problems and Exercises......Page 308
7.1 Introduction......Page 312
7.2 Expectation of Sums of Random Variables......Page 313
7.2.1 Obtaining Bounds from Expectations via the Probabilistic Method......Page 326
7.2.2 The Maximum–Minimums Identity......Page 328
7.3 Moments of the Number of Events that Occur......Page 330
7.4 Covariance, Variance of Sums, and Correlations......Page 337
7.5.1 Definitions......Page 346
7.5.2 Computing Expectations by Conditioning......Page 348
7.5.3 Computing Probabilities by Conditioning......Page 359
7.5.4 Conditional Variance......Page 362
7.6 Conditional Expectation and Prediction......Page 364
7.7 Moment Generating Functions......Page 369
7.7.1 Joint Moment Generating Functions......Page 378
7.8.1 The Multivariate Normal Distribution......Page 380
7.8.2 The Joint Distribution of the Sample Mean and Sample Variance......Page 382
7.9 General Definition of Expectation......Page 384
Summary......Page 385
Problems......Page 388
Theoretical Exercises......Page 395
Self-Test Problems and Exercises......Page 399
8.2 Chebyshev's Inequality and the Weak Law of Large Numbers......Page 403
8.3 The Central Limit Theorem......Page 406
8.4 The Strong Law of Large Numbers......Page 415
8.5 Other Inequalities......Page 418
8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson Random Variable......Page 425
Problems......Page 427
Theoretical Exercises......Page 429
Self-Test Problems and Exercises......Page 430
9.1 The Poisson Process......Page 432
9.2 Markov Chains......Page 434
9.3 Surprise, Uncertainty, and Entropy......Page 440
9.4 Coding Theory and Entropy......Page 443
Summary......Page 449
Problems and Theoretical Exercises......Page 450
References......Page 451
10.1 Introduction......Page 453
10.2 General Techniques for Simulating Continuous Random Variables......Page 455
10.2.1 The Inverse Transformation Method......Page 456
10.2.2 The Rejection Method......Page 457
10.3 Simulating from Discrete Distributions......Page 462
10.4 Variance Reduction Techniques......Page 464
10.4.1 Use of Antithetic Variables......Page 465
10.4.2 Variance Reduction by Conditioning......Page 466
10.4.3 Control Variates......Page 467
Problems......Page 468
Reference......Page 470
Answers to Selected Problems......Page 472
Solutions to Self-Test Problems and Exercises......Page 476
C......Page 536
D......Page 537
E......Page 538
I......Page 539
J......Page 540
N......Page 541
Q......Page 542
S......Page 543
V......Page 544
Z......Page 545