First Course in Probability, A

by Ross, Sheldon

• ISBN13: 9780131856622
• ISBN10: 0131856626
• Edition: 7th
• Format: Hardcover
• Copyright: 2010-01-01
• Publisher: Prentice Hall

Summary

This introduction presents the mathematical theory of probability for readers in the fields of engineering and the sciences who possess knowledge of elementary calculus.Presents new examples and exercises throughout. Offers a new section that presents an elegant way of computing the moments of random variables defined as the number of events that occur. Gives applications to binomial, hypergeometric, and negative hypergeometric random variables, as well as random variables resulting from coupon collecting and match models. Provides additional results on inclusion-exclusion identity, Poisson paradigm, multinomial distribution, and bivariate normal distributionA useful reference for engineering and science professionals.

Table of Contents

Preface	vii
Combinatorial Analysis	1	(23)
Introduction	1	(1)
The Basic Principle of Counting	2	(1)
Permutations	3	(3)
Combinations	6	(4)
Multinomial Coefficients	10	(2)
The Number of Integer Solutions of Equations*	12	(12)
Summary	15	(1)
Problems	16	(3)
Theoretical Exercises	19	(3)
Self-Test Problems and Exercises	22	(2)
Axioms of Probability	24	(42)
Introduction	24	(1)
Sample Space and Events	24	(5)
Axioms of Probability	29	(2)
Some Simple Propositions	31	(6)
Sample Spaces Having Equally Likely Outcomes	37	(12)
Probability as a Continuous Set Function*	49	(4)
Probability as a Measure of Belief	53	(13)
Summary	54	(1)
Problems	55	(6)
Theoretical Exercises	61	(2)
Self-Test Problems and Exercises	63	(3)
Conditional Probability and Independence	66	(66)
Introduction	66	(1)
Conditional Probabilities	66	(6)
Bayes' Formula	72	(15)
Independent Events	87	(14)
P(.|F) Is a Probability	101	(31)
Summary	110	(1)
Problems	111	(13)
Theoretical Exercises	124	(4)
Self-Test Problems and Exercises	128	(4)
Random Variables	132	(73)
Random Variables	132	(6)
Discrete Random Variables	138	(2)
Expected Value	140	(4)
Expectation of a Function of a Random Variable	144	(4)
Variance	148	(2)
The Bernoulli and Binomial Random Variables	150	(10)
Properties of Binomial Random Variables	155	(3)
Computing the Binomial Distribution Function	158	(2)
The Poisson Random Variable	160	(13)
Computing the Poisson Distribution Function	173	(1)
Other Discrete Probability Distributions	173	(10)
The Geometric Random Variable	173	(2)
The Negative Binomial Random Variable	175	(3)
The Hypergeometric Random Variable	178	(4)
The Zeta (or Zipf) Distribution	182	(1)
Properties of the Cumulative Distribution Function	183	(22)
Summary	185	(2)
Problems	187	(10)
Theoretical Exercises	197	(4)
Self-Test Problems and Exercises	201	(4)
Continuous Random Variables	205	(53)
Introduction	205	(4)
Expectation and Variance of Continuous Random Variables	209	(5)
The Uniform Random Variable	214	(4)
Normal Random Variables	218	(12)
The Normal Approximation to the Binomial Distribution	225	(5)
Exponential Random Variables	230	(7)
Hazard Rate Functions	234	(3)
Other Continuous Distributions	237	(5)
The Gamma Distribution	237	(2)
The Weibull Distribution	239	(1)
The Cauchy Distribution	239	(1)
The Beta Distribution	240	(2)
The Distribution of a Function of a Random Variable	242	(16)
Summary	244	(3)
Problems	247	(4)
Theoretical Exercises	251	(3)
Self-Test Problems and Exercises	254	(4)
Jointly Distributed Random Variables	258	(69)
Joint Distribution Functions	258	(9)
Independent Random Variables	267	(13)
Sums of Independent Random Variables	280	(8)
Conditional Distributions: Discrete Case	288	(3)
Conditional Distributions: Continuous Case	291	(5)
Order Statistics*	296	(4)
Joint Probability Distribution of Functions of Random Variables	300	(8)
Exchangeable Random Variables*	308	(19)
Summary	311	(2)
Problems	313	(6)
Theoretical Exercises	319	(4)
Self-Test Problems and Exercises	323	(4)
Properties of Expectation	327	(103)
Introduction	327	(1)
Expectation of Sums of Random Variables	328	(19)
Obtaining Bounds from Expectations via the Probabilistic Method*	342	(2)
The Maximum-Minimums Identity*	344	(3)
Moments of the Number of Events that Occur	347	(8)
Covariance, Variance of Sums, and Correlations	355	(10)
Conditional Expectation	365	(17)
Definitions	365	(2)
Computing Expectations by Conditioning	367	(9)
Computing Probabilities by Conditioning	376	(4)
Conditional Variance	380	(2)
Conditional Expectation and Prediction	382	(5)
Moment Generating Functions	387	(12)
Joint Moment Generating Functions	397	(2)
Additional Properties of Normal Random Variables	399	(5)
The Multivariate Normal Distribution	399	(3)
The Joint Distribution of the Sample Mean and Sample Variance	402	(2)
General Definition of Expectation	404	(26)
Summary	405	(3)
Problems	408	(10)
Theoretical Exercises	418	(8)
Self-Test Problems and Exercises	426	(4)
Limit Theorems	430	(33)
Introduction	430	(1)
Chebyshev's Inequality and the Weak Law of Large Numbers	430	(4)
The Central Limit Theorem	434	(9)
The Strong Law of Large Numbers	443	(2)
Other Inequalities	445	(9)
Bounding The Error Probability	454	(9)
Summary	456	(1)
Problems	457	(2)
Theoretical Exercises	459	(2)
Self-Test Problems and Exercises	461	(2)
Additional Topics in Probability	463	(24)
The Poisson Process	463	(3)
Markov Chains	466	(6)
Surprise, Uncertainty, and Entropy	472	(4)
Coding Theory and Entropy	476	(11)
Summary	483	(1)
Theoretical Exercises	484	(1)
Self-Test Problems and Exercises	485	(2)
Simulation	487	(74)
Introduction	487	(3)
General Techniques for Simulating Continuous Random Variables	490	(7)
The Inverse Transformation Method	490	(1)
The Rejection Method	491	(6)
Simulating from Discrete Distributions	497	(2)
Variance Reduction Techniques	499	(9)
Use of Antithetic Variables	500	(1)
Variance Reduction by Conditioning	501	(2)
Control Variates	503	(1)
Summary	503	(1)
Problems	504	(2)
Self-Test Problems and Exercises	506	(2)
APPENDICES
A Answers to Selected Problems	508	(3)
B Solutions to Self-Test Problems and Exercises	511	(50)
Index	561