A First Course in Probability2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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1 Combinatorial Analysis1

1.1 Introduction1

1.2 The Basic Principle of Counting1

1.3 Permutations3

1.4 Combinations5

1.5 Multinomial Coefficients9

1.6 The Number of Integer Solutions of Equations12

Summary15

Problems16

Theoretical Exercises18

Self-Test Problems and Exercises20

2 Axioms of Probability22

2.1 Introduction22

2.2 Sample Space and Events22

2.3 Axioms of Probability26

2.4 Some Simple Propositions29

2.5 Sample Spaces Having Equally Likely Outcomes33

2.6 Probability as a Continuous Set Function44

2.7 Probability as a Measure of Belief48

Summary49

Problems50

Theoretical Exercises54

Self-Test Problems and Exercises56

3 Conditional Probability and Independence58

3.1 Introduction58

3.2 Conditional Probabilities58

3.3 Bayes's Formula65

3.4 Independent Events79

3.5 P(.F) Is a Probability93

Summary101

Problems102

Theoretical Exercises110

Self-Test Problems and Exercises114

4 Random Variables117

4.1 Random Variables117

4.2 Discrete Random Variables123

4.3 Expected Value125

4.4 Expectation of a Function of a Random Variable128

4.5 Variance132

4.6 The Bernoulli and Binomial Random Variables134

4.6.1 Properties of Binomial Random Variables139

4.6.2 Computing the Binomial Distribution Function142

4.7 The Poisson Random Variable143

4.7.1 Computing the Poisson Distribution Function154

4.8 Other Discrete Probability Distributions155

4.8.1 The Geometric Random Variable155

4.8.2 The Negative Binomial Random Variable157

4.8.3 The Hypergeometric Random Variable160

4.8.4 The Zeta (or Zipf) Distribution163

4.9 Expected Value of Sums of Random Variables164

4.10 Properties of the Cumulative Distribution Function168

Summary170

Problems172

Theoretical Exercises179

Self-Test Problems and Exercises183

5 Continuous Random Variables186

5.1 Introduction186

5.2 Expectation and Variance of Continuous Random Variables190

5.3 The Uniform Random Variable194

5.4 Normal Random Variables198

5.4.1 The Normal Approximation to the Binomial Distribution204

5.5 Exponential Random Variables208

5.5.1 Hazard Rate Functions212

5.6 Other Continuous Distributions215

5.6.1 The Gamma Distribution215

5.6.2 The Weibull Distribution216

5.6.3 The Cauchy Distribution217

5.6.4 The Beta Distribution218

5.7 The Distribution of a Function of a Random Variable219

Summary222

Problems224

Theoretical Exercises227

Self-Test Problems and Exercises229

6 Jointly Distributed Random Variables232

6.1 Joint Distribution Functions232

6.2 Independent Random Variables240

6.3 Sums of Independent Random Variables252

6.3.1 Identically Distributed Uniform Random Variables252

6.3.2 Gamma Random Variables254

6.3.3 Normal Random Variables256

6.3.4 Poisson and Binomial Random Variables259

6.3.5 Geometric Random Variables260

6.4 Conditional Distributions: Discrete Case263

6.5 Conditional Distributions: Continuous Case266

6.6 Order Statistics270

6.7 Joint Probability Distribution of Functions of Random Variables274

6.8 Exchangeable Random Variables282

Summary285

Problems287

Theoretical Exercises291

Self-Test Problems and Exercises293

7 Properties of Expectation297

7.1 Introduction297

7.2 Expectation of Sums of Random Variables298

7.2.1 Obtaining Bounds from Expectations via the Probabilistic Method311

7.2.2 The Maximum-Minimums Identity313

7.3 Moments of the Number of Events that Occur315

7.4 Covariance, Variance of Sums, and Correlations322

7.5 Conditional Expectation331

7.5.1 Definitions331

7.5.2 Computing Expectations by Conditioning333

7.5.3 Computing Probabilities by Conditioning344

7.5.4 Conditional Variance347

7.6 Conditional Expectation and Prediction349

7.7 Moment Generating Functions354

7.7.1 Joint Moment Generating Functions363

7.8 Additional Properties of Normal Random Variables365

7.8.1 The Multivariate Normal Distribution365

7.8.2 The Joint Distribution of the Sample Mean and Sample Variance367

7.9 General Definition of Expectation369

Summary370

Problems373

Theoretical Exercises380

Self-Test Problems and Exercises384

8 Limit Theorems388

8.1 Introduction388

8.2 Chebyshev's Inequality and the Weak Law of Large Numbers388

8.3 The Central Limit Theorem391

8.4 The Strong Law of Large Numbers400

8.5 Other Inequalities403

8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson Random Variable410

Summary412

Problems412

Theoretical Exercises414

Self-Test Problems and Exercises415

9 Additional Topics in Probability417

9.1 The Poisson Process417

9.2 Markov Chains419

9.3 Surprise, Uncertainty, and Entropy425

9.4 Coding Theory and Entropy428

Summary434

Problems and Theoretical Exercises435

Self-Test Problems and Exercises436

References436

10 Simulation438

10.1 Introduction438

10.2 General Techniques for Simulating Continuous Random Variables440

10.2.1 The Inverse Transformation Method441

10.2.2 The Rejection Method442

10.3 Simulating from Discrete Distributions447

10.4 Variance Reduction Techniques449

10.4.1 Use of Antithetic Variables450

10.4.2 Variance Reduction by Conditioning451

10.4.3 Control Variates452

Summary453

Problems453

Self-Test Problems and Exercises455

Reference455

Answers to Selected Problems457

Solutions to Self-Test Problems and Exercises461

Index521