A first course in probability

First Course in Probability, A - Sheldon M. Ross (0137463146) ......Page 1
Contents ......Page 5
Preface ......Page 11
1.1 Introduction ......Page 15
1.2 The Basic Principle of Counting ......Page 16
1.3 Permutations ......Page 17
1.4 Combinations ......Page 19
1.5 Multinomial Coefficients ......Page 24
1.6 On the Distribution of Balls in Urns ......Page 26
Summary ......Page 29
Problems ......Page 30
Theoretical Exercises ......Page 33
Self-Test Problems and Exercises ......Page 37
2.2 Sample Space and Events ......Page 39
2.3 Axioms of Probability ......Page 44
2.4 Some Simple Propositions ......Page 46
2.5 Sample Spaces Having Equally Likely Outcomes ......Page 50
2.6 Probability As a Continuous Set Function ......Page 62
2.7 Probability As a Measure of Belief ......Page 66
Summary ......Page 67
Problems ......Page 68
Theoretical Exercises ......Page 75
Self-Test Problems and Exercises ......Page 78
3.2 Conditional Probabilities ......Page 81
3.3 Bayes' Formula ......Page 86
3.4 Independent Events ......Page 97
3.5 P(- | F) is a Probability ......Page 110
Summary ......Page 117
Problems ......Page 118
Theoretical Exercises ......Page 132
Self-Test Problems and Exercises ......Page 137
4.1 Random Variables ......Page 140
4.2 Distribution Functions ......Page 145
4.3 Discrete Random Variables ......Page 148
4.4 Expected Value ......Page 150
4.5 Expectation of a Function of a Random Variable ......Page 153
4.6 Variance ......Page 156
4.7 The Bernoulli and Binomial Random Variables ......Page 158
4.7.1 Properties of Binomial Random Variables ......Page 163
4.7.2 Computing the Binomial Distribution Function ......Page 166
4.8 The Poisson Random Variable ......Page 168
4.8.1 Computing the Poisson Distribution Function ......Page 175
4.9.1 The Geometric Random Variable ......Page 176
4.9.2 The Negative Binomial Random Variable ......Page 178
4.9.3 The Hypergeometric Random Variable ......Page 181
4.9.4 The Zeta (or Zipf) distribution ......Page 184
Summary ......Page 185
Problems ......Page 187
Theoretical Exercises ......Page 198
Self-Test Problems and Exercises ......Page 203
14.2 Laplace Transforms ......Page 206
5.2 Expectation and Variance of Continuous Random Variables ......Page 209
5.3 The Uniform Random Variable ......Page 214
5.4 Normal Random Variables ......Page 218
5.4.1 The Normal Approximation to the Binomial Distribution ......Page 226
5.5 Exponential Random Variables ......Page 229
5.5.1 Hazard Rate Functions ......Page 234
5.6.1 The Gamma Distribution ......Page 236
5.6.2 The WeibullDistribution ......Page 238
5.6.3 The Cauchy Distribution ......Page 239
5.6.4 The Beta Distribution ......Page 240
5.7 The Distribution of a Function of a Random Variable ......Page 241
Summary ......Page 244
Problems ......Page 246
Theoretical Exercises ......Page 251
Self-Test Problems and Exercises ......Page 255
6.1 Joint Distribution Functions ......Page 258
6.2 Independent Random Variables ......Page 266
6.3 Sums of Independent Random Variables ......Page 278
6.4 Conditional Distributions: Discrete Case ......Page 286
6.5 Conditional Distributions: Continuous Case ......Page 287
6.6 Order Statistics ......Page 290
6.7 Joint Probability Distribution of Functions of Random Variables ......Page 294
6.8 Exchangeable Random Variables ......Page 302
Summary ......Page 305
Problems ......Page 307
Theoretical Exercises ......Page 314
Self-Test Problem and Exercises ......Page 319
7.1 Introduction ......Page 323
7.2 Expectation of Sums of Random Variables ......Page 324
7.3 Covariance, Variance of Sums, and Correlations ......Page 339
7.4.1 Definitions ......Page 349
7.4.2 Computing Expectations by Conditioning ......Page 351
7.4.3 Computing Probabilities by Conditioning ......Page 358
7.4.4 Conditional Variance ......Page 362
7.5 Conditional Expectation and Prediction ......Page 364
7.6 Moment Generating Functions ......Page 369
7.6.1 Joint Moment Generating Functions ......Page 378
7.7.1 The Multivariate Normal Distribution ......Page 379
7.7.2 The Joint Distribution of the Sample Mean and Sample Variance ......Page 380
7.8 General Definition of Expectation ......Page 382
Summary ......Page 384
Problems ......Page 386
Theoretical Exercises ......Page 398
Self-Test Problems and Exercises ......Page 406
8.2 Chebyshev's Inequality and the Weak Law of Large Numbers ......Page 409
8.3 The Central Limit Theorem ......Page 413
8.4 The Strong Law of Large Numbers ......Page 421
8.5 Other Inequalities ......Page 426
8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson ......Page 432
Problems ......Page 435
Theoretical Exercises ......Page 438
Self-Test Problems and Exercises ......Page 440
14.1 Differential Equations ......Page 205
14.3 Difference Equations ......Page 211
14.4 Z Transforms ......Page 213
9.1 The Poisspn Process ......Page 442
9.2 Markov Chains ......Page 445
9.3 Surprise, Uncertainty, and Entropy ......Page 450
Summary ......Page 455
Theoretical Exercises and Problems ......Page 462
References ......Page 464
10.1 Introduction ......Page 466
10.2.1 The Inverse Transformation Method ......Page 469
10.2.2 The Rejection Method ......Page 470
10.3 Simulating from Discrete Distributions ......Page 476
10.4 Variance Reduction Techniques ......Page 478
10.4.1 Use of Antithetic Variables ......Page 479
10.4.2 Variance Reduction by Conditioning ......Page 480
Summary ......Page 482
Problems ......Page 483
References ......Page 486
Appendix A ANSWERS TO SELECTED PROBLEMS ......Page 487
Appendix B SOLUTIONS TO SELF-TEST PROBLEMS AND EXERCISES ......Page 491
INDEX ......Page 527