A First Course in Probability

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1 A First Course in Probability Seventh Edition Sheldon Ross University of Southern California PEARSON Prentice Hall Upper Saddle River, New Jersey 07458

2 Preface 1 Combinatorial Analysis Introduction The Basic Principle of Counting Permutations Combinations Multinomial Coefficients The Number of Integer Solutions of Equations* 12 Summary 15 Problems 16 Theoretical Exercises 19 Seif-Test Problems and Exercises 22 2 Axioms of Probability Introduction Sample Space and Events Axioms of Probability Some Simple Propositions Sample Spaces Having Equally Likely Outcomes Probability as a Continuous Set Function* Probability as a Measure of Belief 53 Summary 54 Problems 55 Theoretical Exercises 61 Self-Test Problems and Exercises 63 vii iii

3 IV 3 Conditional Probability and Independence Introduction Conditional Probabilities Bayes' Formula Independent Events P(- F)Isa Probability 101 Summary 110 Problems 111 Theoretical Exercises 124 Self-Test Problems and Exercises Random Variables Random Variables Discrete Random Variables Expected Value Expectation of a Function of a Random Variable Variance The Bernoulli and Binomial Random Variables Properties of Binomial Random Variables Computing the Binomial Distribution Function The Poisson Random Variable Computing the Poisson Distribution Function Other Discrete Probability Distributions The Geometrie Random Variable The Negative Binomial Random Variable The Hypergeometric Random Variable The Zeta (or Zipf) Distribution Properties of the Cumulative Distribution Function 183 Summary 185 Problems 187 Theoretical Exercises 197 Self-Test Problems and Exercises Continuous Random Variables Introduction Expectation and Variance of Continuous Random Variables The Uniform Random Variable Normal Random Variables The Normal Approximation to the Binomial Distribution Exponential Random Variables Hazard Rate Functions Other Continuous Distributions The Gamma Distribution The Weibull Distribution The Cauchy Distribution The Beta Distribution 240

4 v 5.7 The Distribution of a Function of a Random Variable 242 Summary 244 Problems 247 Theoretical Exercises 251 Self-Test Problems and Exercises Jointly Distributed Random Variables Joint Distribution Functions Independent Random Variables Sums of Independent Random Variables Conditional Distributions: Discrete Case Conditional Distributions: Continuous Case Order Statistics* Joint Probability Distribution of Functions of Random Variables Exchangeable Random Variables* 308 Summary 311 Problems 313 Theoretical Exercises 319 Self-Test Problems and Exercises Properties of Expectation Introduction Expectation of Sums of Random Variables Obtaining Bounds from Expectations via the Probabilistic Method* The Maximum-Minimums Identity* Moments of the Number of Events that Occur Covariance, Variance of Sums, and Correlations Conditional Expectation Definitions Computing Expectations by Conditioning Computing Probabilities by Conditioning Conditional Variance Conditional Expectation and Prediction Moment Generating Functions Joint Moment Generating Functions Additional Properties of Normal Random Variables The Multivariate Normal Distribution The Joint Distribution of the Sample Mean and Sample Variance General Definition of Expectation 404 Summary 405 Problems 408 Theoretical Exercises 418 Self-Test Problems and Exercises 426

5 VI 8 Limit Theorems Introduction Chebyshev's Inequality and the Weak Law of Large Numbers The Central Limit Theorem The Strong Law of Large Numbers Other Inequalities Bounding The Error Probability 454 Summary 456 Problems 457 Theoretical Exercises 459 Seif-Test Problems and Exercises Additional Topics in Probability The Poisson Process Markov Chains Surprise, Uncertainty, and Entropy Coding Theory and Entropy 476 Summary 483 Theoretical Exercises 484 Seif-Test Problems and Exercises Simulation Introduction General Techniques for Simulating Continuous Random Variables The Inverse Transformation Method The Rejection Method Simulating from Discrete Distributions Variance Reduction Techniques Use of Antithetic Variables Variance Reduction by Conditioning Control Variates 503 Summary 503 Problems 504 Self-Test Problems and Exercises 506 APPENDICES A Answers to Selected Problems 508 B Solutions to Self-Test Problems and Exercises 511 Index 561

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