First Course in Probability, A

by Ross, Sheldon M.

• ISBN13: 9780130338518
• ISBN10: 0130338516
• Edition: 6th
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: PRENTICE HALL

Summary

For upper level or undergraduate/graduate level introduction to probability for math, science, engineering, and business students with a background in elementary calculus. This market-leading introduction to probability features exceptionally clear explanations of the mathematics of probability theory and explores its many diverse applications through numerous interesting and motivational examples. The outstanding problem sets and intuitive explanations are hallmark features of this market leading text.

Table of Contents

Preface	vi
Combinatorial Analysis	1	(23)
Introduction	1	(1)
The Basic Principle of Counting	2	(1)
Permutations	3	(2)
Combinations	5	(5)
Multinomial Coefficients	10	(2)
The Number of Integer Solutions of Equations*	12	(12)
Summary	15	(1)
Problems	15	(7)
Theoretical Exercises	22	(1)
Self-Test Problems and Exercises	22	(2)
Axioms of Probability	24	(40)
Introduction	24	(1)
Sample Space and Events	24	(4)
Axioms of Probability	28	(3)
Some Simple Propositions	31	(4)
Sample Spaces Having Equally Likely Outcomes	35	(12)
Probability As a Continuous Set Function*	47	(4)
Probability As a Measure of Belief	51	(13)
Summary	52	(1)
Problems	53	(6)
Theoretical Exercises	59	(2)
Self-Test Problems and Exercises	61	(3)
Conditional Probability and Independence	64	(58)
Introduction	64	(1)
Conditional Probabilities	64	(5)
Bayes' Formula	69	(14)
Independent Events	83	(13)
P(•/F) is a Probability*	96	(26)
Summary	103	(1)
Problems	104	(11)
Theoretical Exercises	115	(4)
Self-Test Problems and Exercises	119	(3)
Random Variables	122	(65)
Random Variables	122	(5)
Discrete Random Variables	127	(3)
Expected Value	130	(3)
Expectation of a Function of a Random Variable	133	(4)
Variance	137	(2)
The Bernoulli and Binomial Random Variables	139	(10)
Properties of Binomial Random Variables	144	(3)
Computing the Binomial Distribution Function	147	(2)
The Poisson Random Variable	149	(9)
Computing the Poisson Distribution Function	157	(1)
Other Discrete Probability Distribution	158	(8)
The Geometric Random Variable	158	(2)
The Negative Binomial Random Variable	160	(2)
The Hypergeometric Random Variable	162	(4)
The Zeta (or Zipf) distribution	166	(1)
Properties of the Cumulative Distribution Function	166	(21)
Summary	169	(2)
Problems	171	(9)
Theoretical Exercises	180	(4)
Self-Test Problems and Exercises	184	(3)
Continuous Random Variables	187	(52)
Introduction	187	(3)
Expectation and Variance of Continuous Random Variables	190	(5)
The Uniform Random Variable	195	(4)
Normal Random Variables	199	(11)
The Normal Approximation to the Binomial Distribution	206	(4)
Exponential Random Variables	210	(7)
Hazard Rate Functions	215	(2)
Other Continuous Distributions	217	(6)
The Gamma Distribution	217	(3)
The Weibull Distribution	220	(1)
The Cauchy Distribution	220	(1)
The Beta Distribution	221	(2)
The Distribution of a Function of a Random Variable	223	(16)
Summary	225	(3)
Problems	228	(4)
Theoretical Exercises	232	(3)
Self-Test Problems and Exercises	235	(4)
Jointly Distributed Random Variables	239	(65)
Joint Distribution Functions	239	(9)
Independent Random Variables	248	(12)
Sums of Independent Random Variables	260	(8)
Conditional Distributions: Discrete Case	268	(2)
Conditional Distributions: Continuous Case	270	(3)
Order Statistics*	273	(4)
Joint Probability Distribution of Functions of Random Variables	277	(8)
Exchangeable Random Variables*	285	(19)
Summary	288	(2)
Problems	290	(6)
Theoretical Exercises	296	(3)
Self-Test Problems and Exercises	299	(5)
Properties of Expectations	304	(96)
Introduction	304	(1)
Expectation of Sums of Random Variables	305	(22)
Obtaining Bounds from Expectations via the Probabilistic Method*	321	(3)
The Maximum-Minimums Identity*	324	(3)
Covariance, Variance of Sums, and Correlations	327	(13)
Conditional Expectation	340	(16)
Definitions	340	(3)
Computing Expectations by Conditioning	343	(7)
Computing Probabilities by Conditioning	350	(4)
Conditional Variance	354	(2)
Conditional Expectation and Prediction	356	(5)
Moment Generating Functions	361	(12)
Joint Moment Generating Functions	371	(2)
Additional Properties of Normal Random Variables	373	(2)
The Multivariate Normal Distribution	373	(1)
The Joint Distribution of the Sample Mean and Sample Variance	374	(1)
General Definition of Expectation*	375	(25)
Summary	377	(2)
Problems	379	(10)
Theoretical Exercises	389	(8)
Self-Test Problems and Exercises	397	(3)
Limit Theorems	400	(32)
Introduction	400	(1)
Chebyshev's Inequality and the Weak Law of Large Numbers	400	(3)
The Central Limit Theorem	403	(9)
The Strong Law of Large Numbers	412	(5)
Other Inequalities	417	(7)
Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson	424	(8)
Summary	426	(1)
Problems	427	(2)
Theoretical Exercises	429	(1)
Self-Test Problems and Exercises	430	(2)
Additional Topics in Probability	432	(23)
The Poisson Process	432	(3)
Markov Chains	435	(5)
Surprise, Uncertainty, and Entropy	440	(5)
Coding Theory and Entropy	445	(10)
Summary	451	(1)
Theoretical Exercises and Problems	452	(2)
Self-Test Problems and Exercises	454	(1)
References	454	(1)
Simulation	455	(20)
Introduction	455	(3)
General Techniques for Simulating Continuous Random Variables	458	(7)
The Inverse Transformation Method	458	(1)
The Rejection Method	459	(6)
Simulating from Discrete Distributions	465	(2)
Variance Reduction Techniques	467	(8)
Use of Antithetic Variables	468	(1)
Variance Reduction by Conditioning	468	(2)
Control Variates	470	(1)
Summary	471	(1)
Problems	471	(3)
Self-Test Problems and Exercises	474	(1)
References	474	(1)
Appendix A Answers to Selected Problems	475	(3)
Appendix B Solutions to Self-Test Problems and Exercises	478	(41)
Index	519