Elements of Applied Probability : For Engineering, Mathematics and Systems Science

by McDonald, David

• ISBN13: 9789812387394
• ISBN10: 9812387390
• Format: Hardcover
• Copyright: 2004-03-01
• Publisher: World Scientific Pub Co Inc

Table of Contents

Introduction	1	(31)
Telecommunications and Quality Control	1	(1)
The Digital Network	1	(5)
Quality Control	6	(8)
Exercises	14	(3)
A Probability Primer	17	(4)
A Random Sample	21	(4)
Joint Distributions	25	(3)
Exercises	28	(3)
Modelling a Stochastic Process	31	(40)
The Probability Space	31	(7)
Random Variables	38	(6)
Expectation	44	(4)
Conditional Probabilities	48	(4)
Independence	52	(3)
General Random Variables	55	(2)
Independence of General Random Variables	57	(2)
The Kolmogorov Extension Theorem	59	(5)
Exercises	64	(7)
Variables and Limit Theorems	71	(28)
Discrete Random Variables	71	(5)
Continuous Random Variables	76	(6)
Generating Functions	82	(3)
Law of Large Numbers	85	(7)
Exercises	92	(7)
The Poisson Process	99	(28)
Introduction	99	(1)
Bernoulli Point Processes	100	(4)
The Poisson Process	104	(3)
Applications of the Poisson Process	107	(3)
A Further Characterization	110	(3)
The Poisson Approximation	113	(6)
Nonhomogeneous Poisson Processes	119	(4)
Exercises	123	(4)
Markov Chains	127	(56)
Introduction	127	(8)
Steady-State Markov Chains	135	(5)
Convergence to Steady-State	140	(7)
Reversible Markov Chains	147	(3)
Hitting Probabilities	150	(11)
Proofs of Convergence to Steady State	161	(4)
Transience and Recurrence	165	(4)
Existence and the Strong Markov Property	169	(2)
The Watched Process	171	(4)
Foster's Criterion for Stability	175	(2)
Exercises	177	(6)
Renewal Theory	183	(46)
Introduction	183	(4)
Renewal Reward Processes	187	(5)
Stationary Renewal Processes	192	(11)
Convergence to Stationarity	203	(6)
Renewal with Nonlattice Variables	209	(1)
Homogeneous Coupling	210	(4)
The Bernoulli Part and Coupling	214	(3)
Proof of the Coupling Theorem	217	(4)
Regenerative Processes	221	(4)
Exercises	225	(4)
Markov Processes	229	(42)
Introduction	229	(2)
The Generator of a Markov Process	231	(11)
Queues and Time Reversal	242	(8)
Jackson Networks	250	(4)
Kelly or BCMP Networks	254	(7)
Approximation by a Markov Process	261	(4)
Exercises	265	(6)
Markov Decision Theory	271	(48)
Introduction	271	(5)
The Optimality Equation	276	(10)
Infinite Horizon Decision Problems	286	(9)
Continuous Time Decision Problems	295	(3)
Optimal Stopping	298	(10)
Quality Control - a Worst Case Analysis	308	(6)
Exercises	314	(5)
Appendix	319	(40)
Useful Results in Analysis	319	(1)
Integration and Expectation	319	(3)
Convexity	322	(2)
Norms and Fixed Points	324	(3)
Algebraic Results	327	(1)
Further Reading	328	(2)
Bibliography	330	(4)
Solutions to Selected Problems	334	(25)
Solutions to Selected Exercises in Chapter 1	334	(2)
Solutions to Selected Exercises in Chapter 2	336	(2)
Solutions to Selected Exercises in Chapter 3	338	(4)
Solutions to Selected Exercises in Chapter 4	342	(4)
Solutions to Selected Exercises in Chapter 5	346	(3)
Solutions to Selected Exercises in Chapter 6	349	(3)
Solutions to Selected Exercises in Chapter 7	352	(2)
Solutions to Selected Exercises in Chapter 8	354	(5)
Index	359