本书通过大量的例子系统介绍了概率论的基础知识及其广泛应用，内容涉及组合分析、条件概率、离散型随机变量、连续型随机变量、随机变量的联合分布、期望的性质、极限定理和模拟等。各章末附有大量的练习，还在书末给出自检习题的全部解答。

1　组合分析　1
1.1　引言　1
1.2　计数基本法则　2
1.3　排列　3
1.4　组合　5
1.5　多项式系数　9
1.6　方程的整数解个数　12
总结　15
问题　15
习题　18
自检习题　20
2　概率论公理　22
2.1　引言　22
2.2　样本空间和事件　22
2.3　概率论公理　26
2.4　几个简单命题　29
2.5　等可能结果的样本空间　33
2.6　概率：连续集函数　44
2.7　概率：确信程度的度量　48
总结　49
问题　50
习题　55
自检习题　56
3　条件概率和独立性　58
3.1　引言　58
3.2　条件概率　58
3.3　贝叶斯公式　64
3.4　独立事件　78
3.5　P(|F)是概率　95
总结　102
问题　103
习题　113
自检习题　116
4　随机变量　119
4.1　随机变量　119
4.2　离散型随机变量　123
4.3　期望　126
4.4　随机变量函数的期望　128
4.5　方差　132
4.6　伯努利随机变量和二项随机变量　137
4.6.1　二项随机变量的性质　142
4.6.2　计算二项分布函数　145
4.7　泊松随机变量　146
4.8　其他离散型概率分布　158
4.8.1　几何随机变量　158
4.8.2　负二项随机变量　160
4.8.3　超几何随机变量　163
4.8.4　ζ分布　167
4.9　随机变量和的期望　167
4.10　累积分布函数的性质　172
总结　174
问题　175
习题　182
自检习题　86
5　连续型随机变量　189
5.1　引言　189
5.2　连续型随机变量的期望和方差　193
5.3　均匀随机变量　197
5.4　正态随机变量　200
5.5　指数随机变量　211
5.6　其他连续型概率分布　218
5.6.1　Γ分布　218
5.6.2　韦布尔分布　219
5.6.3　柯西分布　220
5.6.4　分布　221
5.6.5　帕雷托分布　223
5.7　随机变量函数的分布　224
总结　227
问题　228
习题　231
自检习题　233
6　随机变量的联合分布　237
6.1　联合分布函数　237
6.2　独立随机变量　247
6.3　独立随机变量的和　258
6.3.1　独立同分布均匀随机变量　258
6.3.2　Г随机变量　260
6.3.3　正态随机变量　262
6.3.4　泊松随机变量和二项随机变量　266
6.4　离散情形下的条件分布　267
6.5　连续情形下的条件分布　270
6.6　次序统计量　276
6.7　随机变量函数的联合分布　280
6.8　可交换随机变量　287
总结　290
问题　291
习题　296
自检习题　299
7　期望的性质　303
7.1　引言　303
7.2　随机变量和的期望　304
7.2.1　通过概率方法将期望值作为界　317
7.2.2　关于最大值与最小值的恒等式　319
7.3　试验序列中事件发生次数的矩　321
7.4　随机变量和的协方差、方差及相关系数　328
7.5　条件期望　337
7.5.1　定义　337
7.5.2　通过取条件计算期望　339
7.5.3　通过取条件计算概率　349
7.5.4　条件方差　354
7.6　条件期望及预测　356
7.7　矩母函数　360
7.8　正态随机变量的更多性质　371
7.8.1　多元正态分布　371
7.8.2　样本均值与样本方差的联合分布　373
7.9　期望的一般定义　375
总结　377
问题　378
习题　385
自检习题　390
8　极限定理　394
8.1　引言　394
8.2　切比雪夫不等式及弱大数定律　394
8.3　中心极限定理　397
8.4　强大数定律　406
8.5　其他不等式　409
8.6　用泊松随机变量逼近独立的伯努利随机变量和的概率误差界　418
8.7　洛伦兹曲线　420
总结　424
问题　424
习题　426
自检习题　428
9　概率论的其他课题　430
9.1　泊松过程　430
9.2　马尔可夫链　432
9.3　惊奇、不确定性及熵　437
9.4　编码定理及熵　441
总结　447
习题　447
自检习题　448
10　模拟　450
10.1　引言　450
10.2　模拟连续型随机变量的一般方法　453
10.2.1　逆变换方法　453
10.2.2　舍取法　454
10.3　模拟离散分布　459
10.4　方差缩减技术　462
10.4.1　利用对偶变量　463
10.4.2　利用“条件”　463
10.4.3　控制变量　465
总结　465
问题　466
自检习题　467
部分习题答案　468
自检习题解答　470
索引　502
离散型分布　506
连续型分布　508


CONTENTS

1 COMBINATORIAL ANALYSIS 1
1.1 Introduction 1
1.2 TheBasic Principle of Counting 2
1.3 Permutations 3
1.4 Combinations 5
1.5 Multinomial Coef.cients 9
1.6 The Number of Integer Solutions of Equations 12
Summary 15
Problems 15
Theoretical Exercises 18
Self-Test Problems and Exercises 20
2 AXIOMSOF 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 Probabilityasa Continuous Set Function 44
2.7 Probabilityasa Measure of Belief 48
Summary 49
Problems 50
Theoretical Exercises 55
Self-Test Problems and Exercises 56
3 CONDITIONAL PROBABILITY AND INDEPENDENCE 58
3.1 Introduction 58
3.2 Conditional Probabilities 58
3.3 Bayes’sFormula 64
3.4 Independent Events 78
3.5 P(·|F)Is a Probability 95
Summary 102
Problems 103
Theoretical Exercises 113
Self-Test Problems and Exercises 116
4 RANDOM VARIABLES 119
4.1 Random Variables 119
4.2 Discrete RandomVariables 123
4.3 Expected Value 126
4.4 Expectation of a Function of a Random Variable 128
4.5 Variance 132
4.6 The Bernoulli and Binomial Random Variables 137
4.6.1 Properties of Binomial Random Variables 142
4.6.2 Computing the Binomial Distribution Function 145
4.7 The Poisson Random Variable 146
4.7.1 Computing the Poisson Distribution Function 158
4.8 Other Discrete Probability Distributions 158
4.8.1 The Geometric Random Variable 158
4.8.2 The Negative Binomial Random Variable 160
4.8.3 The Hypergeometric Random Variable 163
4.8.4 TheZeta(or Zipf)Distribution 167
4.9 Expected Value of Sums of Random Variables 167
4.10 Properties of the Cumulative Distribution Function 172
Summary 174
Problems 175
Theoretical Exercises 182
Self-Test Problems and Exercises 186
5 CONTINUOUS RANDOM VARIABLES 189
5.1 Introduction 189
5.2 Expectation and Variance of Continuous Random Variables 193
5.3 The Uniform Random Variable 197
5.4 Normal Random Variables 200
5.4.1 The Normal Approximation to the Binomial Distribution 207
5.5 Exponential Random Variables 211
5.5.1 Hazard Rate Functions 215
5.6 Other Continuous Distributions 218
5.6.1 The Gamma Distribution 218
5.6.2 The Weibull Distribution 219
5.6.3 The Cauchy Distribution 220
5.6.4 The Beta Distribution 221
5.6.5 The Pare to Distribution 223
5.7 The Distribution of a Function of a Random Variable 224
Summary 227
Problems 228
Theoretical Exercises 231
Self-Test Problems and Exercises 233
6 JOINTLY DISTRIBUTED RANDOM VARIABLES 237
6.1 Joint Distribution Functions 237
6.2 Independent Random Variables 247
6.3 Sums of Independent Random Variables 258
6.3.1 Identically Distributed Uniform Random Variables 258
6.3.2 Gamma Random Variables 260
6.3.3 Normal Random Variables 262
6.3.4 Poissonand Binomial Random Variables 266
6.4 Conditional Distributions: Discrete Case 267
6.5 Conditional Distributions: Continuous Case 270
6.6 Order Statistics 276
6.7 Joint Probability Distribution of Functions of Random Variables 280
6.8 Exchangeable Random Variables 287
Summary 290
Problems 291
Theoretical Exercises 296
Self-Test Problems and Exercises 299
7 PROPERTIES OF EXPECTATION 303
7.1 Introduction 303
7.2 Expectation of Sums of Random Variables 304
7.2.1 Obtaining Bounds from Expectationsvia the Probabilistic Method 317
7.2.2 The Maximum–Minimums Identity 319
7.3 Moments of the Number of Events that Occur 321
7.4 Covariance,Variance of Sums, and
Correlations 328

7.5 Conditional Expectation 337
7.5.1 De.nitions 337
7.5.2 Computing Expectations by Conditioning 339
7.5.3 Computing Probabilities by Conditioning 349
7.5.4 Conditional Variance 354
7.6 Conditional Expectation and Prediction 356
7.7 Moment Generating Functions 360
7.7.1 Joint Moment Generating Functions 369
7.8 Additional Properties of Normal Random Variables 371
7.8.1 The Multivariate Normal Distribution 371
7.8.2 The Joint Distribution of the Sample Mean and SampleVariance 373
7.9 GeneralDe.nitionof Expectation 375
Summary 377
Problems 378
Theoretical Exercises 385
Self-Test Problems and Exercises 390
8 LIMIT THEOREMS 394
8.1 Introduction 394
8.2 Chebyshev’s Inequality and the Weak Law of Large Numbers 394
8.3 The Central Limit Theorem 397
8.4 The Strong Law of Large Numbers 406
8.5 Other Inequalities and a PoissonLimit Result 409
8.6 Bounding the Error Probability When Approximating a Sum of Independent
Bernoulli Random Variables by a Poisson Random Variable 418
8.7 The Lorenz Curve 420
Summary 424
Problems 424
Theoretical Exercises 426
Self-Test Problems and Exercises 428
9 ADDITIONAL TOPICS IN PROBABILITY 430
9.1 The Poisson Process 430
9.2 Markov Chains 432
9.3 Surprise, Uncertainty, and Entropy 437
9.4 Coding TheoryandEntropy 441
Summary 447
Problems and Theoretical Exercises 447
Self-Test Problems and Exercises 448
10 SIMULATION 450
10.1 Introduction 450
10.2 General Techniques for Simulating
Continuous Random Variables 453

10.2.1 The Inverse Transformation Method 453
10.2.2 The Rejection Method 454
10.3 Simulating from Discrete Distributions 459
10.4 Variance Reduction Techniques 462
10.4.1 Useof Antithetic Variables 463
10.4.2 Variance Reductionby Conditioning 463
10.4.3 Control Variates 465
Summary 465
Problems 466
Self-Test Problems and Exercises 467
Answers to Selected Problems 468
Solutions to Self-Test Problems and Exercises 470
Index 502
Common Discrete Distributions 506
Common Continuous Distributions 508