Table of Contents

Chapter 1Introduction to Probability1

1.1Random Experiments2

1.2Sample Space2

1.3Relations and Operations between Events3

1.4The Definition of Probability7

1.5Equally Likely Outcomes Model10

1.6Conditional Probability17

1.7Total Probability and Bayes' Theorem20

1.8Independent Events24

Exercise 126

Chapter 2Random Variables and Distributions31

2.1Random Variables32

2.2Cumulative Distribution Function33

2.3Discrete Distributions35

2.4Some Common Discrete Distributions35

2.5Continuous Distributions41

2.6Some Useful Continuous Distributions43

2.7Functions of a Random Variable50

Exercise 256

Chapter 3Multivariate Probability Distributions61

3.1Bivariate Distributions62

3.2Marginal Distributions70

3.3Conditional Distributions73

3.4Independent Random Variables81

3.5Functions of Two or More Random Variables86

Exercise 3106

Chapter 4Characteristics of Random Variables113

4.1The Expectation of a Random Variable114

4.2Variance122

4.3The Characteristics of Some Common Distributions124

4.4Chebyshevs Inequality130

4.5Covariance and Correlation Coefficient131

4.6Moment and Covariance Matrix139

Exercise 4143

Chapter 5Large Random Samples149

5.1The Law of Large Numbers150

5.2The Central Limit Theorem152

Exercise 5156

Chapter 6Estimation159

6.1Population and Sample160

6.2Moment Estimation163

6.3Maximum Likelihood Estimation165

6.4Properties of Estimators170

6.5Three Important Distributions172

6.6Confidence Intervals182

Exercise 6191

Chapter 7Hypothesis Testing197

7.1Basics of Hypothesis Testing198

7.2Hypothesis Tests for a Population Mean200

7.3Testing Differences between Means207

7.4Hypothesis Tests for One or Two Variances210

7.5Goodness of Fit Tests214

Exercise 7219

Chapter 8Linear Regression225

8.1Linear Regression Model226

8.2Least Squares Estimation227

8.3Properties of Linear Regression Estimators230

8.4Inferences Concerning the Slope234

8.5Regression Validity236

8.6Confidence Interval for Mean Response237

8.7Inference for Prediction239

Exercise 8242

Chapter 9Introduction to R Language245

9.1Features of R Language246

9.2R Installation247

9.3Vector, Matrix and Data Frame248

9.4Loop and Branch Control Statements252

9.5Common Probability Distributions255

9.6Some Examples256

Appendix ABinomial Probability Distribution262

Appendix BPoisson Cumulative Distribution265

Appendix CStandard Normal Table268

Appendix Dtdistribution Upper Quantiles tα(n)270

Appendix Eχ2distribution Upper Quantiles χ2α(n)273

Appendix FFdistribution Upper Quantiles Fα(n1,n2)276

Appendix GSome Common Probability Distributions281

Bibliography283

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