Introduction to Probability: Multivariate Models and Applications

1 Two-dimensional discrete random variables 1

1.1 Introduction 2

1.2 Joint probability function 2

1.3 Marginal distributions 16

1.4 Expectation of a function 25

1.5 Conditional distributions and expectations 33

1.6 Basic concepts and formulas 44

1.7 Computational exercises 45

1.8 Self-assessment exercises 50

1.8.1 True { False Questions 50

1.8.2 Multiple Choice Questions 51

1.9 Review Problems 54

1.10 Applications 59

1.10.1 Mixture distributions and reinsurance 59

2 Two-dimensional continuous random variables 65

2.1 Introduction 66

2.2 Joint density function 66

2.3 Marginal distributions 78

2.4 Expectation of a function 85

2.5 Conditional distributions and expectations 88

2.6 Geometric probability 98

2.7 Basic concepts and formulas 105

2.8 Computational exercises 106

2.9 Self-assessment exercises 113

2.9.1 True { False Questions 113

2.9.2 Multiple Choice Questions 115

2.10 Review problems 118

2.11 Applications 120

2.11.1 Modeling proportions 120

3 Independence and multivariate distributions 127

3.1 Introduction 128

3.2 Independence 128

3.3 Properties of independent random variables 144

3.4 Multivariate joint distributions 150

3.5 Independence of more than two variables 164

3.6 Distribution of an ordered sample 175

3.7 Basic concepts and formulas 187

3.8 Computational exercises 191

3.9 Self-assessment exercises 198

3.9.1 True { False Questions 198

3.9.2 Multiple Choice Questions 200

3.10 Review Problems 204

3.11 Applications 210

3.11.1 Acceptance sampling 210

4 Transformations of variables 217

4.1 Introduction 218

4.2 Joint distribution for functions of variables 218

4.3 Distributions of sum, di
erence, product, quotient 227

4.4 2, t and F distributions 240

4.5 Basic concepts and formulas 256

4.6 Computational exercises 258

4.7 Self-assessment exercises 263

4.7.1 True { False Questions 263

4.7.2 Multiple Choice Questions 265

4.8 Review problems 268

4.9 Applications 4 272

4.9.1 Random number generators coverage { Planning under

random event occurrences 272

5 Covariance and correlation 279

5.1 Introduction 279

5.2 Covariance 280

5.3 Correlation coe
cient 295

5.4 Conditional expectation and variance 305

5.5 Regression curves 318

5.6 Basic concepts and formulas 333

5.7 Computational exercises 335

5.8 Self-assessment exercises 342

5.8.1 True { False Questions 342

5.8.2 Multiple Choice Questions 344

5.9 Review problems 348

5.10 Applications 355

5.10.1 Portfolio Optimization Theory (Markowitz (1952)) 355

6 Important multivariate distributions 361

6.1 Introduction 362

6.2 Multinomial distribution 362

6.3 Multivariate hypergeometric distribution 375

6.4 Bivariate normal distribution 390

6.5 Basic concepts and formulas 404

6.6 Computational exercises 406

6.7 Self-assessment exercises 413

6.7.1 True { False Questions 413

6.7.2 Multiple Choice Questions 415

6.8 Review problems 418

6.9 Applications 422

6.9.1 The e
ect of dependence on the distribution of the sum 422

7 Generating functions 427

7.1 Introduction 428

7.2 Moment generating function 428

7.3 Moment generating functions of some important distributions 438

7.4 Moment generating functions for sum of variables 445

7.5 Probability generating function 454

7.6 Characteristic function 466

7.7 Generating functions for multivariate case 472

7.8 Basic concepts and formulas 481

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7.9 Computational exercises 483

7.10 Self-assessment exercises 486

7.10.1 True { False Questions 486

7.10.2 Multiple Choice Questions 489

7.11 Review problems 493

7.12 Applications 502

7.12.1 Random Walks 502

8 Limit theorems 511

8.1 Introduction 512

8.2 Laws of large numbers 512

8.3 Central Limit Theorem 521

8.4 Basic concepts and formulas 539

8.5 Computational exercises 541

8.6 Self-assessment exercises 545

8.6.1 True { False Questions 545

8.6.2 Multiple Choice Questions 546

8.7 Review problems 550

8.8 Applications 553

8.8.1 Use of the CLT for capacity planning 553