viii Contents 4.7 The Poisson Random Variable 143 471 ing the Poisson Distribution Function 154 4.8 Other Discrete Probability Distributions. 48 The Geometric Random varable Variable 483 cometric Random Variable 555 48.4 163 170 Problems 1 ercise 13 5 Continuous Random Variables 186 2 190 5.3 The Uniform Random Variable 194 5.4 al Rormal 288 Appro 5.Rate Functions 212 5.6 Other Continuous Distributions 。， 215 。 5.63 The cauchy distribution 217 5.6.4 The Beta Distribution 5.7 The Distribution of a Function of a Random Variable Problems 224 xercises 。 227 229 6 Jointly Distributed Random Variables 232 61 6 ndent Random y 252 6.3.1 Identically Distributed Uniform Random Variables 252 633 Gamma 34 Poormal R Random Variables 25 ial D m Variables 25d 6.3.5 Geometric Random Variables 260 Conditional Distributions:Discrete Case istributions:Continuous Case. 6.7 Joint Probability Distribution of Functions of Random Variables 274 6.8 Exchangeable Random Variables. pummary. Theoretical Exercises Self-Test Problems and Exercises.viii Contents 4.7 The Poisson Random Variable . . . . . . . . . . . . . . . . . . . . . . . 143 4.7.1 Computing the Poisson Distribution Function . . . . . . . . . . 154 4.8 Other Discrete Probability Distributions . . . . . . . . . . . . . . . . . 155 4.8.1 The Geometric Random Variable . . . . . . . . . . . . . . . . . 155 4.8.2 The Negative Binomial Random Variable . . . . . . . . . . . . 157 4.8.3 The Hypergeometric Random Variable . . . . . . . . . . . . . 160 4.8.4 The Zeta (or Zipf) Distribution . . . . . . . . . . . . . . . . . . 163 4.9 Expected Value of Sums of Random Variables . . . . . . . . . . . . . 164 4.10 Properties of the Cumulative Distribution Function . . . . . . . . . . . 168 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . 183 5 Continuous Random Variables 186 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.2 Expectation and Variance of Continuous Random Variables . . . . . 190 5.3 The Uniform Random Variable . . . . . . . . . . . . . . . . . . . . . . 194 5.4 Normal Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.4.1 The Normal Approximation to the Binomial Distribution . . . 204 5.5 Exponential Random Variables . . . . . . . . . . . . . . . . . . . . . . 208 5.5.1 Hazard Rate Functions . . . . . . . . . . . . . . . . . . . . . . . 212 5.6 Other Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . 215 5.6.1 The Gamma Distribution . . . . . . . . . . . . . . . . . . . . . 215 5.6.2 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . 216 5.6.3 The Cauchy Distribution . . . . . . . . . . . . . . . . . . . . . . 217 5.6.4 The Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . 218 5.7 The Distribution of a Function of a Random Variable . . . . . . . . . 219 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . 229 6 Jointly Distributed Random Variables 232 6.1 Joint Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . 232 6.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . 240 6.3 Sums of Independent Random Variables . . . . . . . . . . . . . . . . . 252 6.3.1 Identically Distributed Uniform Random Variables . . . . . . 252 6.3.2 Gamma Random Variables . . . . . . . . . . . . . . . . . . . . 254 6.3.3 Normal Random Variables . . . . . . . . . . . . . . . . . . . . 256 6.3.4 Poisson and Binomial Random Variables . . . . . . . . . . . . 259 6.3.5 Geometric Random Variables . . . . . . . . . . . . . . . . . . . 260 6.4 Conditional Distributions: Discrete Case . . . . . . . . . . . . . . . . . 263 6.5 Conditional Distributions: Continuous Case . . . . . . . . . . . . . . . 266 6.6 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 6.7 Joint Probability Distribution of Functions of Random Variables . . . 274 6.8 Exchangeable Random Variables . . . . . . . . . . . . . . . . . . . . . 282 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Theoretical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Self-Test Problems and Exercises . . . . . . . . . . . . . . . . . . . . . 293