E-Book Overview
text is designed for an introductory probability course at the university level for sophomores, juniors, and seniors in mathematics, physical and social sciences, engineering, and computer science. It presents a thorough treatment of ideas and techniques necessary for a firm understanding of the subject. The text is also recommended for use in discrete probability courses. The material is organized so that the discrete and continuous probability discussions are presented in a separate, but parallel, manner. This organization does not emphasize an overly rigorous or formal view of probabililty and therefore offers some strong pedagogical value. Hence, the discrete discussions can sometimes serve to motivate the more abstract continuous probability discussions. Features: Key ideas are developed in a somewhat leisurely style, providing a variety of interesting applications to probability and showing some nonintuitive ideas. Over 600 exercises provide the opportunity for practicing skills and developing a sound understanding of ideas. Numerous historical comments deal with the development of discrete probability. The text includes many computer programs that illustrate the algorithms or the methods of computation for important problems.
E-Book Content
Introduction to Probability
Charles M. Grinstead Swarthmore College
J. Laurie Snell Dartmouth College
To our wives and in memory of Reese T. Prosser
Contents 1 Discrete Probability Distributions 1.1 Simulation of Discrete Probabilities . . . . . . . . . . . . . . . . . . . 1.2 Discrete Probability Distributions . . . . . . . . . . . . . . . . . . . .
1 1 18
2 Continuous Probability Densities 2.1 Simulation of Continuous Probabilities . . . . . . . . . . . . . . . . . 2.2 Continuous Density Functions . . . . . . . . . . . . . . . . . . . . . .
41 41 55
3 Combinatorics 75 3.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3 Card Shuffling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4 Conditional Probability 4.1 Discrete Conditional Probability . . . . . . . . . . . . . . . . . . . . 4.2 Continuous Conditional Probability . . . . . . . . . . . . . . . . . . . 4.3 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133 133 162 175
5 Distributions and Densities 183 5.1 Important Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.2 Important Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6 Expected Value and Variance 225 6.1 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.2 Variance of Discrete Random Variables . . . . . . . . . . . . . . . . . 257 6.3 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . 268 7 Sums of Random Variables 285 7.1 Sums of Discrete Random Variables . . . . . . . . . . . . . . . . . . 285 7.2 Sums of Continuous Random Variables . . . . . . . . . . . . . . . . . 291 8 Law of Large Numbers 305 8.1 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . 305 8.2 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . 316 v
vi
CONTENTS
9 Central Limit Theorem 9.1 Bernoulli Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Discrete Independent Trials . . . . . . . . . . . . . . . . . . . . . . . 9.3 Continuous Independent Trials . . . . . . . . . . . . . . . . . . . . .
325 325 340 356
10 Generating Functions 365 10.1 Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 365 10.2 Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 10.3 Continuous Densities . . . . . . . . . . . . . . . . . . .