1 COMBINATORIAL ANALYSIS 1
1.1 Introduction 1
1.2 The Basic Principle of Counting 2
1.3 Permutations 3
1.4 Combinations 5
1.5 Multinomial Coefficients 9
1.6 The Number of Integer Solutions of Equations 12
2 AXIOMS OF PROBABILITY 21
2.1 Introduction 21
2.2 Sample Space and Events 21
2.3 Axioms of Probability 25
2.4 Some Simple Propositions 28
2.5 Sample Spaces Having Equally Likely Outcomes 32
2.6 Probability as a Continuous Set Function 42
2.7 Probability as a Measure of Belief 46
3 CONDITIONAL PROBABILITY AND INDEPENDENCE 56
3.1 Introduction 56
3.2 Conditional Probabilities 56
3.3 Bayes's Formula 62
3.4 Independent Events 75
3.5 P（·｜F）Is a Probability 89
4 RANDOM VARIABLES 112
4.1 Random Variables 112
4.2 Discrete Random Variables 116
4.3 Expected Value 119
4.4 Expectation of a Function of a Random Variable 121
4.5 Variance 125
4.6 The Bernoulli and Binomial Random Variables 127
4.7 The Poisson Random Variable 135
4.8 Other Discrete Probability Distributions 147
4.9 Expected Value of Sums of Random Variables 155
4.10 Properties of the Cumulative Distribution Function 159
5 CONTINUOUS RANDOM VARIABLES 176
5.1 Introduction 176
5.2 Expectation and Variance of Continuous Random Variables 179
5.3 The Uniform Random Variable 184
5.4 Normal Random Variables 187
5.5 Exponential Random Variables 197
5.6 Other Continuous Distributions 203
5.7 The Distribution of a Function of a Random Variable 208
6 JOINTLY DISTRIBUTED RANDOM VARIABLES 220
6.1 Joint Distribution Functions 220
6.2 Independent Random Variables 228
6.3 Sums of Independent Random Variables 239
6.4 Conditional Distributions：Discrete Case 248
6.5 Conditional Distributions：Continuous Case 250
6.6 Order Statistics 256
6.7 Joint Probability Distribution of Functions of Random Variables 260
6.8 Exchangeable Random Variables 267
7 PROPERTIES OF EXPECTATION 280
7.1 Introduction 280
7.2 Expectation of Sums of Random Variables 281
7.3 Moments of the Number of Events that Occur 298
7.4 Covariance,Variance of Sums.and Correlations 304
7.5 Conditional Expectation 313
7.6 Conditional Expectation and Prediction 330
7.7 Moment Generating Functions 334
7.8 Additional Properties of Normal Random Variables 345
7.9 General Definition of Expectation 349
8 LIMIT THEOREMS 367
8.1 Introduction 367
8.2 Chebyshev's Inequality and the Weak Law of Large Numbers 367
8.3 The Central Limit Theorem 370
8.4 The Strong Law of Large Numbers 378
8.5 Other Inequalities 382
8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson Random Variable 388
9 ADDITIONAL TOPICS IN PROBABILITY 395
9.1 The Poisson Process 395
9.2 Markov Chains 397
9.3 Surprise,Uncertainty,and Entropy 402
9.4 Coding Theory and Entropy 405
10 SIMULATION 415
10.1 Introduction 415
10.2 General Techniques for Simulating Continuous Random Variables 417
10.3 Simulating from Discrete Distributions 424
10.4 Variance Reduction Techniques 426
Answers to Selected Problems 433
Solutions to Self-Test Problems and Exercises 435
Index 465