trestles / book-intro-to-probability-2e

1 Sample Space and Probability p1

• 1.1 Sets p3
• 1.2 Probabilistic Models p6
• 1.3 Conditional Probability p18
• 1.4 Total Probability: Theorem and Bayes' Rule p28
• 1.5 Independence p34
• 1.6 Counting p44
• 1.7 Summary and Discussion p51
• Problems p53

2 Discrete Random Variables p71

• 2.1 Basic Concepts p72
• 2.2 Probability Mass Function p74
• 2.3 Functions of Random Variables p80
• 2.4 Expectation, Mean, and Variance p81
• 2.5 Joint PMF's of Multiple Random Variables p92
• 2.6 Conditioning p97
• 2.7 Independence p109
• 2.8 Summary and Discussion p115
• Problems p119

3 General Random Variables p139

• 3.1 Continuous Random Variables and PDFs p140
• 3.2 Cumulative Distribution Functions p148
• 3.3 Normal Random Variables p153
• 3.4 Joint PDFs of Multiple Random Variables p158
• 3.5 Conditioning p164
• 3.6 The Continuous Bayes' Rule p178
• 3.7 Summary and Discussion p182
• Problems

4 Futher Topics on Random Variables p201

• 4.1 Derived Distributions p202
• 4.2 Covariance and Correlation p217
• 4.3 Conditional Expectation and Variance Revisited p222
• 4.4 Transforms p229
• 4.5 Sum of a Random Number of Independent Random Variables p240
• 4.6 Summary and Discussion p244
• Problems p246

5 Limit Theorems p263

• 5.1 Markov and Chebyshev Inequalities p265
• 5.2 The Weak Law of Large Numbers p269
• 5.3 Convergence in Probability p271
• 5.4 The Central Limit Theorem p273
• 5.5 The Strong Law of Large Numbers p280
• 5.6 Summary and Discussion p282
• Problems

6. The Bernoulli and Poisson Processes p295

• 6.1 The Bernoulli Process p297
• 6.2 The Poisson Process p309
• 6.3 Summary and Discussion p324
• Problems p326

7. Markov Chains p339

• 7.1 Discrete-Time Markov Chains p340
• 7.2 Classification of States p346
• 7.3 Steady-State Behavior p352
• 7.4 Absorption Probabilities and Expected Time to Absorption p362
• 7.5 Continuous-Time Markov Chains p369
• 7.6 Summary and Discussion p378
• Problems p380

8 Bayesian Statistical Inference p407

• 8.1 Bayesian Inference and the Posterior Distribution p412
• 8.2 Point Estimation, Hypothesis Testing, and the MAP Rule p420
• 8.3 Bayesian Least Mean Squares Estimation p430
• 8.4 Bayesian Linear Least Mean Squares Estimation p437
• 8.5 Summary and Discussion p444
• Problems p446

9. Classical Statistical Inference p459

• 9.1 Classic Parameter Estimation p462
• 9.2 Linear Regression p477
• 9.3 Binary Hypothesis Testing p486
• 9.4 Significance Testing p496
• 9.5 Summary and Discussion p505
• Problems p507

Index p521