公 CONTENTS 5 Discrete Distributions 107 5.1 Discrete Random Variables ............. 107 5.2 The Discrete Uniform Distribution 110 5.3 The Binomial Distribution 111 5.4 Expectation and Moment Generating Functions... 116 5.5 The Empirical Distribution.·················· 120 5.6 Other Discrete Distributions 123 5.7 Functions of Discrete Random Variables 130 Chapter Exercises........................... 132 6 Continuous Distributions 137 6.1 Continuous Random Variables..... 。 137 6.2 The Continuous Uniform Distribution 142 6.3 The Normal Distribution.. 143 6.4 Functions of Continuous Random Variables 146 6.5 Other Continuous Distributions...... 150 Chapter Exercises.············ 155 7 Multivariate Distributions 157 7.1 Joint and Marginal Probability Distributions.·····. 157 7.2 Joint and Marginal Expectation .. 163 7.3 Conditional Distributions .. 165 7.4 Independent Random Variables... 167 7.5 Exchangeable Random Variables 170 7.6 The Bivariate Normal Distribution 170 7.7 Bivariate Transformations of Random Variables......... 172 7.8 Remarks for the Multivariate Case 175 7.9 The Multinomial Distribution 178 Chapter Exercises..········ 180 8 Sampling Distributions 181 8.1 Simple Random Samples ...... 。。g 182 8.2 Sampling from a Normal Distribution 182 8.3 The Central Limit Theorem....... 185 8.4 Sampling Distributions of Two-Sample Statistics 187 8.5 Simulated Sampling Distributions 189 Chapter Exercises............... 191 9 Estimation 193 9.1 Point Estimation... 193 9.2 Confidence Intervals for Means....,············· 202 9.3 Confidence Intervals for Differences of Means.··..·...·· 208 9.4 Confidence Intervals for Proportions 210 9.5 Confidence Intervals for Variances 212 9.6 Fitting Distributions....... 212 9.7 Sample Size and Margin of Error................ 212 9.8 Other Topics.....·············· 214 Chapter Exercises ............ 215iv CONTENTS 5 Discrete Distributions 107 5.1 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 The Discrete Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.4 Expectation and Moment Generating Functions . . . . . . . . . . . . . . . . . 116 5.5 The Empirical Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.6 Other Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.7 Functions of Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . 130 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6 Continuous Distributions 137 6.1 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 The Continuous Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . 142 6.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.4 Functions of Continuous Random Variables . . . . . . . . . . . . . . . . . . . 146 6.5 Other Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7 Multivariate Distributions 157 7.1 Joint and Marginal Probability Distributions . . . . . . . . . . . . . . . . . . . 157 7.2 Joint and Marginal Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.3 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.4 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.5 Exchangeable Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.6 The Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . 170 7.7 Bivariate Transformations of Random Variables . . . . . . . . . . . . . . . . . 172 7.8 Remarks for the Multivariate Case . . . . . . . . . . . . . . . . . . . . . . . . 175 7.9 The Multinomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8 Sampling Distributions 181 8.1 Simple Random Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 8.2 Sampling from a Normal Distribution . . . . . . . . . . . . . . . . . . . . . . 182 8.3 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.4 Sampling Distributions of Two-Sample Statistics . . . . . . . . . . . . . . . . 187 8.5 Simulated Sampling Distributions . . . . . . . . . . . . . . . . . . . . . . . . 189 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9 Estimation 193 9.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.2 Confidence Intervals for Means . . . . . . . . . . . . . . . . . . . . . . . . . . 202 9.3 Confidence Intervals for Differences of Means . . . . . . . . . . . . . . . . . . 208 9.4 Confidence Intervals for Proportions . . . . . . . . . . . . . . . . . . . . . . . 210 9.5 Confidence Intervals for Variances . . . . . . . . . . . . . . . . . . . . . . . . 212 9.6 Fitting Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 9.7 Sample Size and Margin of Error . . . . . . . . . . . . . . . . . . . . . . . . . 212 9.8 Other Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215