Page 18 - Probability and Statistical Inference

Contents

Preface v

Acknowledgments xi
1 Notions of Probability 1
1.1 Introduction 1
1.2 About Sets 3
1.3 Axiomatic Development of Probability 6
1.4 The Conditional Probability and Independent Events 9
1.4.1 Calculus of Probability 12
1.4.2 Bayes’s Theorem 14
1.4.3 Selected Counting Rules 16
1.5 Discrete Random Variables 18
1.5.1 Probability Mass and Distribution Functions 19
1.6 Continuous Random Variables 23
1.6.1 Probability Density and Distribution Functions 23
1.6.2 The Median of a Distribution 28
1.6.3 Selected Reviews from Mathematics 28
1.7 Some Standard Probability Distributions 32
1.7.1 Discrete Distributions 33
1.7.2 Continuous Distributions 37
1.8 Exercises and Complements 50

2 Expectations of Functions of Random Variables 65
2.1 Introduction 65
2.2 Expectation and Variance 65
2.2.1 The Bernoulli Distribution 71
2.2.2 The Binomial Distribution 72
2.2.3 The Poisson Distribution 73
2.2.4 The Uniform Distribution 73
2.2.5 The Normal Distribution 73
2.2.6 The Laplace Distribution 76
2.2.7 The Gamma Distribution 76
2.3 The Moments and Moment Generating Function 77
2.3.1 The Binomial Distribution 80
2.3.2 The Poisson Distribution 81
2.3.3 The Normal Distribution 82