Concepts in Probability and Stochastic Modeling

by Higgins, James J.; Keller-McNulty, Sallie

• ISBN13: 9780534231361
• ISBN10: 0534231365
• Format: Hardcover
• Copyright: 1994-09-01
• Publisher: Cengage Learning

Summary

This text stresses modern ideas, including simulation and interpretation of results. It focuses on the aspects of probability most relevant to applications, such as stochastic modeling, Markov chains, reliability, and queuing.

Table of Contents

Basic Probability	1	(42)
Sample Spaces and Events	2	(4)
Assignment of Probabilities	6	(8)
Simulation of Events on the Computer	14	(6)
Counting Techniques	20	(7)
Conditional Probability	27	(8)
Independent Events	35	(8)
Discrete Random Variables	43	(57)
Random Variables	43	(8)
Joint Distributions and Independent Random Variables	51	(7)
Expected Values	58	(6)
Variance and Standard Deviation	64	(7)
Sampling and Simulation	71	(7)
Sample Statistics	78	(4)
Expected Values of Jointly Distributed Random Variables and the Law of Large Numbers	82	(8)
Covariance and Correlation	90	(5)
Conditional Expected Values	95	(5)
Special Discrete Random Variables	100	(36)
Binomial Random Variable	100	(11)
Geometric and Negative Binomial Random Variables	111	(5)
Hypergeometric Random Variable	116	(4)
Multinomial Random Variables	120	(4)
Poisson Random Variable	124	(5)
Moments and Moment-Generating Functions	129	(7)
Markov Chains	136	(53)
Introduction: Modeling a Simple Queuing System	136	(4)
The Markov Property	140	(9)
Computing Probabilities for Markov Chains	149	(8)
The Simple Queuing System Revisited	157	(4)
Simulating the Behavior of a Markov Chain	161	(7)
Steady-State Probabilities	168	(9)
Absorbing States and First Passage Times	177	(12)
Continuous Random Variables	189	(47)
Probability Density Functions	189	(12)
Expected Value and Distribution of a Function of a Random Variable	201	(10)
Simulating Continuous Random Variables	211	(8)
Joint Probability Distributions	219	(17)
Special Continuous Random Variables	236	(32)
Exponential Random Variable	236	(5)
Normal Random Variable	241	(9)
Gamma Random Variable	250	(5)
The Weibull Random Variable	255	(4)
Moment-Generating Functions	259	(3)
Method of Moments Estimation	262	(6)
Markov Counting and Queuing Processes	268	(56)
Bernoulli Counting Process	268	(8)
The Poisson Process	276	(4)
Exponential Random Variables and the Poisson Process	280	(3)
The Bernoulli Single-Server Queuing Process	283	(9)
The M/M/1 Queuing Process	292	(7)
k-Server Queuing Process	299	(7)
Balance Equations and Steady-State Probabilities	306	(9)
More Markov Queuing Processes	315	(3)
Simulating an M/M/k Queuing Process	318	(6)
The Distribution of Sums of Random Variables	324	(21)
Sums of Random Variables	324	(5)
Sums of Random Variables and the Central Limit Theorem	329	(7)
Confidence Intervals for Means	336	(5)
A Random Sum of Random Variables	341	(4)
Selected Systems Models	345	(28)
Distribution of Extremes	345	(5)
Scheduling Problems	350	(6)
Sojourns and Transitions for Continuous-Time Markov Processes	356	(7)
Sojourns and Transitions for Continuous-Time Semi-Markov Processes	363	(6)
Sojourns and Transitions for Queuing Process	369	(4)
Reliability Models	373	(28)
The Reliability Function	373	(8)
Hazard Rate	381	(5)
Renewal Processes	386	(6)
Maintained Systems	392	(5)
Nonhomogeneous Poisson Process and Reliability Growth	397	(4)
Appendix	401	(9)
Table 1 Unit Random Numbers	402	(2)
Table 2 Binomial Cumulative Distribution Function	404	(3)
Table 3 Poisson Cumulative Distribution Function	407	(1)
Table 4 Standard Normal Cumulative Distribution Function	408	(1)
Table 5 Incomplete Gamma Function	409	(1)
References	410	(1)
Answers to Selected Odd-Numbered Exercises	411	(6)
Index	417