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Chapter 1 Probability 1

1.1 Introduction 1

1.2 Sample Space and Events 1

1.3 Axioms of Probability 5

1.4 Some Simple Propositions 6

1.5 Sample Spaces Having Equally Likely Outcomes 9

1.6 Conditional Probabilities 10

1.7 Bayes's Formula 13

1.8 Independent Events 17

Exercise 1 20

Chapter 2 Discrete Random Variables and Probability Distributions 22

2.1 Random Variables 22

2.2 Probability Distributions for Discrete Random Variables 24

2.3 The Binomial and Poisson Probability Distributions 29

Exercise 2 34

Chapter 3 Continuous Random Variables and Probability Distributions 37

3.1 Probability Density Functions 37

3.2 Cumulative Distribution Functions 38

3.3 The Normal Distributions 41

3.4 The Exponential and Gamma Distributions 44

3.5 Other Continuous Distributions 48

Exercise 3 48

Chapter 4 Joint Probability Distributions 51

4.1 Joint Distribution Functions 51

4.2 Marginal Distribution Functions 54

4.3 Conditional Distributions 57

4.4 Independent Random Variables 60

4.5 Sums of Independent Random Variables 63

4.6 Joint Probability Distribution of Function of Random Variables 66

Exercise 4 67

Chapter 5 Mathematical Expectation 70

5.1 Expectation of a Random Variable 70

5.2 Variance and Variance of sums 75

5.3 Covariance and Correlations 78

5.4 Conditional Expectation 81

5.5 Chebyshev's Theorem 84

Exercise 5 86

Chapter 6 Limit Theorems 89

6.1 The Weak Law of Large Numbers 89

6.2 The Central Limit Theorem 91

Exercise 6 94

Chapter 7 Descriptive Statistics 96

7.1 Populations and Samples 96

7.2 Measures of Center and Variability 98

7.3 Sampling Distributions 100

7.4 Chi-squared Distribution,t-Distribution and F-Distribution 107

Exercise 7 115

Chapter 8 Estimation 118

8.1 Point Estimation 119

8.2 The Particular Properties of Estimators 123

8.3 Interval Estimation 130

Exercise 8 138

Appendix 140

Reference 156