A First Course in Probability

by Ross, Sheldon M.

• ISBN13: 9780137463145
• ISBN10: 0137463146
• Edition: 5th
• Format: Hardcover
• Copyright: 1997-08-01
• Publisher: PRENTICE HALL

Summary

This market leader is written as an elementary introduction to the mathematical theory of probability for students in mathematics, engineering, and the sciences who possess the prerequisite knowledge of elementary calculus. A major thrust of the Fifth Edition has been to make the book more accessible to today's students. The exercise sets have been revised to include more simple, mechanical problems and a new section of Self-Test Problems with fully worked out solutions conclude each chapter. In addition, many new applications have been added to demonstrate the importance of probability in real situations. A software diskette, referenced in text and packaged with each copy of the book, provides an easy to use tool for students to derive probabilities for binomial, Poisson, and normal random variables, illustrate and explore the central limit theorem, work with the strong law of large numbers, and more.

Table of Contents

PREFACE	xi
1 COMBINATORIAL ANALYSIS	1	(24)
1.1 Introduction	1	(1)
1.2 The Basic Principle of Counting	2	(1)
1.3 Permutations	3	(2)
1.4 Combinations	5	(5)
1.5 Multinomial Coefficients	10	(2)
1.6 On the Distribution of Balls in Urns*	12	(3)
Summary	15	(1)
Problems	16	(3)
Theoretical Exercises	19	(4)
Self-Test Problems and Exercises	23	(2)
2 AXIOMS OF PROBABILITY	25	(42)
2.1 Introduction	25	(1)
2.2 Sample Space and Events	25	(5)
2.3 Axioms of Probability	30	(2)
2.4 Some Simple Propositions	32	(4)
2.5 Sample Spaces Having Equally Likely Outcomes	36	(12)
2.6 Probability As a Continuous Set Function*	48	(4)
2.7 Probability As a Measure of Belief	52	(1)
Summary	53	(1)
Problems	54	(7)
Theoretical Exercises	61	(3)
Self-Test Problems and Exercises	64	(3)
3 CONDITIONAL PROBABILITY AND INDEPENDENCE	67	(59)
3.1 Introduction	67	(1)
3.2 Conditional Probabilities	67	(5)
3.3 Bayes' Formula	72	(11)
3.4 Independent Events	83	(13)
3.5 P(.\F) is a Probability(*)	96	(7)
Summary	103	(1)
Problems	104	(14)
Theoretical Exercises	118	(5)
Self-Test Problems and Exercises	123	(3)
4 RANDOM VARIABLES	126	(66)
4.1 Random Variables	126	(5)
4.2 Distribution Functions	131	(3)
4.3 Discrete Random Variables	134	(2)
4.4 Expected Value	136	(3)
4.5 Expectation of a Function of a Random Variable	139	(3)
4.6 Variance	142	(2)
4.7 The Bernoulli and Binomial Random Variables	144	(10)
4.7.1 Properties of Binomial Random Variables	149	(3)
4.7.2 Computing the Binomial Distribution Function	152	(2)
4.8 The Poisson Random Variable	154	(8)
4.8.1 Computing the Poisson Distribution Function	161	(1)
4.9 Other Discrete Probability Distribution	162	(9)
4.9.1 The Geometric Random Variable	162	(2)
4.9.2 The Negative Binomial Random Variable	164	(3)
4.9.3 The Hypergeometric Random Variable	167	(3)
4.9.4 The Zeta (or Zipf) distribution	170	(1)
Summary	171	(2)
Problems	173	(11)
Theoretical Exercises	184	(5)
Self-Test Problems and Exercises	189	(3)
5 CONTINUOUS RANDOM VARIABLES	192	(52)
5.1 Introduction	192	(3)
5.2 Expectation and Variance of Continuous Random Variables	195	(5)
5.3 The Uniform Random Variable	200	(4)
5.4 Normal Random Variables	204	(11)
5.4.1 The Normal Approximation to the Binomial Distribution	212	(3)
5.5 Exponential Random Variables	215	(7)
5.5.1 Hazard Rate Functions	220	(2)
5.6 Other Continuous Distributions	222	(5)
5.6.1 The Gamma Distribution	222	(2)
5.6.2 The Weibull Distribution	224	(1)
5.6.3 The Cauchy Distribution	225	(1)
5.6.4 The Beta Distribution	226	(1)
5.7 The Distribution of a Function of a Random Variable	227	(3)
Summary	230	(2)
Problems	232	(5)
Theoretical Exercises	237	(4)
Self-Test Problems and Exercises	241	(3)
6 JOINTLY DISTRIBUTED RANDOM VARIABLES	244	(65)
6.1 Joint Distribution Functions	244	(8)
6.2 Independent Random Variables	252	(2)
6.3 Sums of Independent Random Variables	264	(8)
6.4 Conditional Distributions: Discrete Case	272	(1)
6.5 Conditional Distributions: Continuous Case	273	(3)
6.6 Order Statistics*	276	(4)
6.7 Joint Probability Distribution of Functions of Random Variables	280	(8)
6.8 Exchangeable Random Variables*	288	(3)
Summary	291	(2)
Problems	293	(7)
Theoretical Exercises	300	(5)
Self-Test Problem and Exercises	305	(4)
7 PROPERTIES OF EXPECTATION	309	(86)
7.1 Introduction	309	(1)
7.2 Expectation of Sums of Random Variables	310	(15)
7.3 Covariance, Variance of Sums, and Correlations	325	(10)
7.4 Conditional Expectation	335	(15)
7.4.1 Definitions	335	(2)
7.4.2 Computing Expectations by Conditioning	337	(7)
7.4.3 Computing Probabilities by Conditioning	344	(4)
7.4.4 Conditional Variance	348	(2)
7.5 Conditional Expectation and Prediction	350	(5)
7.6 Moment Generating Functions	355	(10)
7.6.1 Joint Moment Generating Functions	364	(1)
7.7 Additional Properties of Normal Random Variables	365	(3)
7.7.1 The Multivariate Normal Distribution	365	(1)
7.7.2 The Joint Distribution of the Sample Mean and Sample Variance	366	(2)
7.8 General Definition of Expectation(*)	368	(2)
Summary	370	(2)
Problems	372	(12)
Theoretical Exercises	384	(8)
Self-Test Problems and Exercises	392	(3)
8 LIMIT THEOREMS	395	(33)
8.1 Introduction	395	(1)
8.2 Chebyshev's Inequality and the Weak Law of Large Numbers	395	(4)
8.3 The Central Limit Theorem	399	(8)
8.4 The Strong Law of Large Numbers	407	(5)
8.5 Other Inequalities	412	(6)
8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson	418	(3)
Summary	421	(1)
Problems	421	(3)
Theoretical Exercises	424	(2)
Self-Test Problems and Exercises	426	(2)
9 ADDITIONAL TOPICS IN PROBABILITY	428	(24)
9.1 The Poisson Process	428	(3)
9.2 Markov Chains	431	(5)
9.3 Surprise, Uncertainty, and Entropy	436	(5)
9.4 Coding Theory and Entropy	441	(1)
Summary	441	(7)
Theoretical Exercises and Problems	448	(2)
Self-Test Problems and Exercises	450	(1)
References	450	(2)
10 SIMULATION	452	(21)
10.1 Introduction	452	(3)
10.2 General Techniques for Simulating Continuous Random Variables	455	(7)
10.2.1 The Inverse Transformation Method	455	(1)
10.2.2 The Rejection Method	456	(6)
10.3 Simulating from Discrete Distributions	462	(2)
10.4 Variance Reduction Techniques	464	(4)
10.4.1 Use of Antithetic Variables	465	(1)
10.4.2 Variance Reduction by Conditioning	466	(2)
10.4.3 Control Variates	468	(1)
Summary	468	(1)
Problems	469	(2)
Self-Test Problems and Exercises	472	(1)
References	472	(1)
Appendix A ANSWERS TO SELECTED PROBLEMS	473	(4)
Appendix B SOLUTIONS TO SELF-TEST PROBLEMS AND EXERCISES	477	(36)
INDEX	513