Fundamentals of Probability, with Stochastic Processes

by Ghahramani, Saeed

• ISBN13: 9780130113290
• ISBN10: 0130113298
• Edition: 3rd
• Format: Hardcover
• Copyright: 2005-01-01
• Publisher: Prentice Hall

Summary

Comprehensive and class-tested, this book is designed for a course in Basic Probability to be taken by mathematics, physics, engineering, statistics, actuarial science, operations research, and computer science majors. It assumes a second course in calculus. The aim of the book is to present probability in the most natural way: through a great number of attractive and instructive examples and exercises that motivate the definitions, theorems, and methodology of the theory. Examples and exercises have been very carefully designed to arouse students' curiosity, motivating them to delve into the theory with enthusiasm. Unique discussions of probability problems published in recent journals are featured to stimulate classroom discussion or individual investigation.

Table of Contents

Preface	xi
Axioms of Probability	1	(37)
Introduction	1	(3)
Sample Space and Events	4	(7)
Axioms of Probability	11	(7)
Basic Theorems	18	(9)
Continuity of Probability Function	27	(2)
Probabilities 0 and 1	29	(2)
Random Selection of Points from Intervals	31	(7)
Review Problems	35	(3)
Combinatorial Methods	38	(37)
Introduction	38	(1)
Counting Principle	38	(9)
Number of Subsets of a Set	42	(1)
Tree Diagrams	43	(4)
Permutations	47	(7)
Combinations	54	(16)
Stirling's Formula	70	(5)
Review Problems	72	(3)
Conditional Probability and Independence	75	(54)
Conditional Probability	75	(10)
Reduction of Sample Space	79	(6)
Law of Multiplication	85	(4)
Law of Total Probability	89	(10)
Bayes' Formula	99	(7)
Independence	106	(23)
Review Problems	126	(3)
Distribution Functions and Discrete Random Variables	129	(45)
Random Variables	129	(5)
Distribution Functions	134	(9)
Discrete Random Variables	143	(7)
Expectations of Discrete Random Variables	150	(11)
Variances and Moments of Discrete Random Variables	161	(9)
Moments	167	(3)
Standardized Random Variables	170	(4)
Review Problems	171	(3)
Special Discrete Distributions	174	(44)
Bernoulli and Binomial Random Variables	174	(14)
Expectations and Variances of Binomial Random Variables	180	(8)
Poisson Random Variable	188	(15)
Poisson as an Approximation to Binomial	188	(6)
Poisson Process	194	(9)
Other Discrete Random Variables	203	(15)
Geometric Random Variable	203	(2)
Negative Binomial Random Variable	205	(2)
Hypergeometric Random Variable	207	(8)
Review Problems	215	(3)
Continuous Random Variables	218	(29)
Probability Density Functions	218	(9)
Density Function of a Function of a Random Variable	227	(6)
Expectations and Variances	233	(14)
Expectations of Continuous Random Variables	233	(6)
Variances of Continuous Random Variables	239	(6)
Review Problems	245	(2)
Special Continuous Distributions	247	(40)
Uniform Random Variable	247	(7)
Normal Random Variable	254	(15)
Correction for Continuity	257	(12)
Exponential Random Variable	269	(7)
Gamma Distribution	276	(5)
Beta Distribution	281	(6)
Review Problems	285	(2)
Joint distributions	287	(80)
Bivariate Distributions	287	(18)
Joint Probability Functions	287	(3)
Joint Probability Density Functions	290	(15)
Independent Random Variables	305	(11)
Conditional Distributions	316	(13)
Multivariate Distributions	329	(16)
Order Statistics	345	(7)
Multinomial Distributions	352	(4)
Transformations of Two Random Variables	356	(11)
Review Problems	362	(5)
More Expectations and Variances	367	(49)
Expected Values of Sums of Random Variables	367	(16)
Pattern Appearance†	376	(7)
Covariance	383	(9)
Correlation	392	(5)
Conditioning on Random Variables	397	(11)
Bivariate Normal Distribution	408	(8)
Review Problems	414	(2)
Sums of Independent Random Variables and Limit Theorems	416	(45)
Moment-Generating Functions	416	(11)
Sums of Independent Random Variables	427	(10)
Markov and Chebyshev Inequalities	437	(6)
Laws of Large Numbers	443	(9)
Central Limit Theorem	452	(9)
Review Problems	458	(3)
Simulation	461	(26)
Introduction	461	(5)
Simulation of Combinatorial Problems	466	(3)
Simulation of Conditional Probabilities	469	(4)
Simulation of Random Variables	473	(8)
Monte Carlo Method	481	(6)
Appendix Tables	487	(4)
Answers to Odd-Numbered Exercises	491	(10)
Index	501