Loading......

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 Solutlons 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 Space 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  58ts
3.2 Conditional Probalgllltle  58
3.3 Bayes's Formula  65
3.4 lnoependent Events  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 Variables123
4.3 Expected Value  125
4.4 Expectation of a Function of a Random Variable  128
4.5 Variance  132
4.6 The Bernoulh and Binomial Random Variables  134
  4.6.1 Properties of Binomial Random Variables  139
  4.6.2 Computing the Binomial Distribution Function  142
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 The Zeta (or Zipf) 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 Exercises183
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 Exponential Random Variables 208
  5.5.1 Hazard Rate Functions  .212
5.6 Other Continuous Distributions  215
  5.6.1 The Gamma Dlstrlbutlon  215
  5.6.2 The Weibull DlStrlbutlon  216
  5.6.3 The-Cauchy Distribution  217
  .6.4 The Beta DlStrlbutlon  218
5.7 The Distribution of a Function of a Random Variable  219
  Summary  222
  Problems  224
  Theoretical Exercises227
  Self-Test Problems and Exercises229
6 Jointly Distributed Random Variables232
6.1 Joint Distribution Functions  232
6.2 Independent Random Variables240
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 Polsson and Binomial Random Variables  259
  6 3 5 Geometric Random Variables  260
  6.4 Conditional Distribution：Discrete Case  263
  6.5 Conditional Distribution：Continuous Case 266
  6 6 Order Statistics  270
  6.7 Joint Probability Distribution of Functions of Random Variables  274
  6.8 Exciaanzeaole Random Variables  282
  Summary  285
  Problems  287
  Theoretical Exercises291
  elf Test Problems and Exercises  293
7 Properties of Expectation  29")
  7.1 Introduction  297
  7.2 Expectation of Sums of Random Variabl 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 ConditionalExpectation  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 ConditionalVariance  347
7.6 Conditional Expectation and Prediction  349
7.7 Moment Generating Functions  354
  7.7.1 Joint Moment Generating Functions  363
  7.8  Addltlona proprietaries of Normal Random Variables 365
  7.8.1 The Multivariate Normal Dlstrlbution 365
  7.8.2 The Joint Distribution of the Sample Mean and Sample Variance  367
7.9 General Definition of Expectation369
  Summary  37(3
  Problems  373
  Theoretical Exercises 38C
  Self-Test Problems and Exercises38d
8 Limit Theorems  388
8.1 Introduction  388
8.2 Chebyshev's Inequality and the Weak Law of Large Numbers  388
8.3 The Central Limit Theorem  391
8.4 The Strong Law of Large Numbers  40C
8.5 Other Inequamles  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 Exercises436
  References  436
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
10.4Variance 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  521