1　組合分析　1
1.1　引言　1
1.2　計數(shù)基本法則　2
1.3　排列　3
1.4　組合　5
1.5　多項式系數(shù)　9
1.6　方程的整數(shù)解個數(shù)　12
總結(jié)　15
問題　15
習(xí)題　18
自檢習(xí)題　20
2　概率論公理　22
2.1　引言　22
2.2　樣本空間和事件　22
2.3　概率論公理　26
2.4　幾個簡單命題　29
2.5　等可能結(jié)果的樣本空間　33
2.6　概率：連續(xù)集函數(shù)　44
2.7　概率：確信程度的度量　48
總結(jié)　49
問題　50
習(xí)題　55
自檢習(xí)題　56
3　條件概率和獨立性　58
3.1　引言　58
3.2　條件概率　58
3.3　貝葉斯公式　64
3.4　獨立事件　78
3.5　P(|F)是概率　95
總結(jié)　102
問題　103
習(xí)題　113
自檢習(xí)題　116
4　隨機變量　119
4.1　隨機變量　119
4.2　離散型隨機變量　123
4.3　期望　126
4.4　隨機變量函數(shù)的期望　128
4.5　方差　132
4.6　伯努利隨機變量和二項隨機變量　137
4.6.1　二項隨機變量的性質(zhì)　142
4.6.2　計算二項分布函數(shù)　145
4.7　泊松隨機變量　146
4.8　其他離散型概率分布　158
4.8.1　幾何隨機變量　158
4.8.2　負二項隨機變量　160
4.8.3　超幾何隨機變量　163
4.8.4　ζ分布　167
4.9　隨機變量和的期望　167
4.10　累積分布函數(shù)的性質(zhì)　172
總結(jié)　174
問題　175
習(xí)題　182
自檢習(xí)題　86
5　連續(xù)型隨機變量　189
5.1　引言　189
5.2　連續(xù)型隨機變量的期望和方差　193
5.3　均勻隨機變量　197
5.4　正態(tài)隨機變量　200
5.5　指數(shù)隨機變量　211
5.6　其他連續(xù)型概率分布　218
5.6.1　Γ分布　218
5.6.2　韋布爾分布　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　隨機變量函數(shù)的分布　224
總結(jié)　227
問題　228
習(xí)題　231
自檢習(xí)題　233
6　隨機變量的聯(lián)合分布　237
6.1　聯(lián)合分布函數(shù)　237
6.2　獨立隨機變量　247
6.3　獨立隨機變量的和　258
6.3.1　獨立同分布均勻隨機變量　258
6.3.2　Г隨機變量　260
6.3.3　正態(tài)隨機變量　262
6.3.4　泊松隨機變量和二項隨機變量　266
6.4　離散情形下的條件分布　267
6.5　連續(xù)情形下的條件分布　270
6.6　次序統(tǒng)計量　276
6.7　隨機變量函數(shù)的聯(lián)合分布　280
6.8　可交換隨機變量　287
總結(jié)　290
問題　291
習(xí)題　296
自檢習(xí)題　299
7　期望的性質(zhì)　303
7.1　引言　303
7.2　隨機變量和的期望　304
7.2.1　通過概率方法將期望值作為界　317
7.2.2　關(guān)于最大值與最小值的恒等式　319
7.3　試驗序列中事件發(fā)生次數(shù)的矩　321
7.4　隨機變量和的協(xié)方差、方差及相關(guān)系數(shù)　328
7.5　條件期望　337
7.5.1　定義　337
7.5.2　通過取條件計算期望　339
7.5.3　通過取條件計算概率　349
7.5.4　條件方差　354
7.6　條件期望及預(yù)測　356
7.7　矩母函數(shù)　360
7.8　正態(tài)隨機變量的更多性質(zhì)　371
7.8.1　多元正態(tài)分布　371
7.8.2　樣本均值與樣本方差的聯(lián)合分布　373
7.9　期望的一般定義　375
總結(jié)　377
問題　378
習(xí)題　385
自檢習(xí)題　390
8　極限定理　394
8.1　引言　394
8.2　切比雪夫不等式及弱大數(shù)定律　394
8.3　中心極限定理　397
8.4　強大數(shù)定律　406
8.5　其他不等式　409
8.6　用泊松隨機變量逼近獨立的伯努利隨機變量和的概率誤差界　418
8.7　洛倫茲曲線　420
總結(jié)　424
問題　424
習(xí)題　426
自檢習(xí)題　428
9　概率論的其他課題　430
9.1　泊松過程　430
9.2　馬爾可夫鏈　432
9.3　驚奇、不確定性及熵　437
9.4　編碼定理及熵　441
總結(jié)　447
習(xí)題　447
自檢習(xí)題　448
10　模擬　450
10.1　引言　450
10.2　模擬連續(xù)型隨機變量的一般方法　453
10.2.1　逆變換方法　453
10.2.2　舍取法　454
10.3　模擬離散分布　459
10.4　方差縮減技術(shù)　462
10.4.1　利用對偶變量　463
10.4.2　利用“條件”　463
10.4.3　控制變量　465
總結(jié)　465
問題　466
自檢習(xí)題　467
部分習(xí)題答案　468
自檢習(xí)題解答　470
索引　502
離散型分布　506
連續(xù)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