Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers

by Roy D. Yates; David J. Goodman

• ISBN13: 9780471178378
• ISBN10: 0471178373
• Format: Hardcover
• Copyright: 1999-01-01
• Publisher: Wiley

Summary

What Does Winning the Lottery Have To do with Engineering? Whether you're trying to win millions in the lottery or designing a complex computer network, you're applying probability theory. Although you encounter probability applications everywhere, the theory can be deceptively difficult to learn and apply correctly. This text will help you grasp the concepts of probability and stochastic processes and apply them throughout your careers. These concepts are clearly presented throughout the book as a sequence of building blocks that are clearly identified as either an axiom, definition, or theorem. This approach provides you with a better understanding of the material which you'll be able to use to solve practical problems. Key Features: * The text follows a single model that begins with an experiment consisting of a procedure and observations. * The mathematics of discrete random variables appears separately from the mathematics of continuous random variables. * Stochastic processes are introduced in Chapter 6, immediately after the presentation of discrete and continuous random variables. Subsequent material, including central limit theorem approximations, laws of large numbers, and statistical inference, then use examples that reinforce stochastic process concepts. * An abundance of exercises are provided that help students learn how to put the theory to use.

Table of Contents

CHAPTER 1 EXPERIMENTS, MODELS, AND PROBABILITIES	1	(42)
Getting Started with Probability	1	(2)
1.1 Set Theory	3	(4)
1.2 Applying Set Theory to Probability	7	(5)
1.3 Probability Axioms	12	(3)
1.4 Some Consequences of the Axioms	15	(1)
1.5 Conditional Probability	16	(5)
1.6 Independence	21	(3)
1.7 Sequential Experiments and Tree Diagrams	24	(3)
1.8 Counting Methods	27	(4)
1.9 Independent Trials	31	(5)
Chapter Summary	36	(1)
Problems	36	(7)
CHAPTER 2 DISCRETE RANDOM VARIABLES	43	(44)
2.1 Definitions	43	(3)
2.2 Probability Mass Function	46	(3)
2.3 Some Useful Discrete Random Variables	49	(6)
2.4 Cumulative Distribution Function (CDF)	55	(4)
2.5 Averages	59	(5)
2.6 Functions of a Random Variable	64	(4)
2.7 Expected Value of a Derived Random Variable	68	(2)
2.8 Variance and Standard Deviation	70	(4)
2.9 Conditional Probability Mass Function	74	(5)
Chapter Summary	79	(1)
Problems	80	(7)
CHAPTER 3 MULTIPLE DISCRETE RANDOM VARIABLES	87	(32)
3.1 Joint Probability Mass Function	87	(3)
3.2 Marginal PMF	90	(3)
3.3 Functions of Two Random Variables	93	(1)
3.4 Expectations	94	(6)
3.5 Conditioning a Joint PMF by an Event	100	(2)
3.6 Conditional PMF	102	(4)
3.7 Independent Random Variables	106	(2)
3.8 More Than Two Discrete Random Variables	108	(3)
Chapter Summary	111	(1)
Problems	112	(7)
CHAPTER 4 CONTINUOUS RANDOM VARIABLES	119	(46)
Continuous Sample Space	119	(2)
4.1 The Cumulative Distribution Function	121	(2)
4.2 Probability Density Function	123	(6)
4.3 Expected Values	129	(3)
4.4 Some Useful Continuous Random Variables	132	(5)
4.5 Gaussian Random Variables	137	(7)
4.6 Delta Functions, Mixed Random Variables	144	(6)
4.7 Probability Models of Derived Random Variables	150	(5)
4.8 Conditioning a Continuous Random Variable	155	(4)
Chapter Summary	159	(1)
Problems	159	(6)
CHAPTER 5 MULTIPLE CONTINUOUS RANDOM VARIABLES	165	(36)
5.1 Joint Cumulative Distribution Function	165	(2)
5.2 Joint Probability Density Function	167	(5)
5.3 Marginal PDF	172	(2)
5.4 Functions of Two Random Variables	174	(3)
5.5 Expected Values	177	(2)
5.6 Conditioning a Joint PDF by an Event	179	(2)
5.7 Conditional PDF	181	(3)
5.8 Independent Random Variables	184	(2)
5.9 Jointly Gaussian Random Variables	186	(5)
5.10 More Than Two Continuous Random Variables	191	(4)
Chapter Summary	195	(1)
Problems	196	(5)
CHAPTER 6 STOCHASTIC PROCESSES	201	(30)
Definitions	201	(2)
6.1 Stochastic Process Examples	203	(2)
6.2 Types of Stochastic Processes	205	(2)
6.3 Random Variables from Random Processes	207	(3)
6.4 Independent, Identically Distributed Random Sequences	210	(1)
6.5 The Poisson Process	211	(4)
6.6 The Brownian Motion Process	215	(1)
6.7 Expected Value and Correlation	216	(3)
6.8 Stationary Processes	219	(4)
6.9 Wide Sense Stationary Random Processes	223	(2)
Chapter Summary	225	(1)
Problems	226	(5)
CHAPTER 7 SUMS OF RANDOM VARIABLES	231	(30)
7.1 Expectations of Sums	231	(4)
7.2 PDF of the Sum of Two Random Variables	235	(1)
7.3 Moment Generating Function	236	(4)
7.4 MGF of the Sum of Independent Random Variables	240	(2)
7.5 Sums of Independent Gaussian Random Variables	242	(2)
7.6 Random Sums of Independent Random Variables	244	(3)
7.7 Central Limit Theorem	247	(5)
7.8 Applications of the Central Limit Theorem	252	(3)
Chapter Summary	255	(1)
Problems	256	(5)
CHAPTER 8 THE SAMPLE MEAN	261	(18)
8.1 Expected Value and Variance	261	(2)
8.2 Useful Inequalities	263	(3)
8.3 Sample Mean of Large Numbers	266	(3)
8.4 Laws of Large Numbers	269	(6)
Chapter Summary	275	(1)
Problems	276	(3)
CHAPTER 9 STATISTICAL INFERENCE	279	(44)
9.1 Significance Testing	281	(2)
9.2 Binary Hypothesis Testing	283	(9)
9.3 Multiple Hypothesis Test	292	(3)
9.4 Estimation of a Random Variable	295	(5)
9.5 Linear Estimation of X given Y	300	(7)
9.6 MAP and ML Estimation	307	(3)
9.7 Estimation of Model Parameters	310	(6)
Chapter Summary	316	(1)
Problems	317	(6)
CHAPTER 10 RANDOM SIGNAL PROCESSING	323	(22)
10.1 Linear Filtering of a Random Process	323	(4)
10.2 Power Spectral Density	327	(3)
10.3 Cross Correlations	330	(5)
10.4 Gaussian Processes	335	(3)
10.5 White Gaussian Noise Processes	338	(2)
10.6 Digital Signal Processing	340	(1)
Chapter Summary	341	(1)
Problems	342	(3)
CHAPTER 11 RENEWAL PROCESSES AND MARKOV CHAINS	345	(52)
11.1 Renewal Processes	345	(6)
11.2 Poisson Process	351	(4)
11.3 Renewal-Reward Processes	355	(2)
11.4 Discrete Time Markov Chains	357	(3)
11.5 Discrete Time Markov Chain Dynamics	360	(3)
11.6 Limiting State Probabilities	363	(4)
11.7 State Classification	367	(6)
11.8 Limit Theorems For Discrete Time Markov Chains	373	(4)
11.9 Periodic States and Multiple Communicating Classes	377	(4)
11.10 Continuous Time Markov Chains	381	(5)
11.11 Birth-Death Processes and Queueing Systems	386	(5)
Chapter Summary	391	(1)
Problems	392	(5)
APPENDIX A COMMON RANDOM VARIABLES	397	(6)
A.1 Discrete Random Variables	397	(2)
A.2 Continuous Random Variables	399	(4)
APPENDIX B QUIZ SOLUTIONS	403	(46)
Quiz Solutions--Chapter 1	403	(4)
Quiz Solutions--Chapter 2	407	(5)
Quiz Solutions--Chapter 3	412	(7)
Quiz Solutions--Chapter 4	419	(4)
Quiz Solutions--Chapter 5	423	(5)
Quiz Solutions--Chapter 6	428	(3)
Quiz Solutions--Chapter 7	431	(4)
Quiz Solutions--Chapter 8	435	(2)
Quiz Solutions--Chapter 9	437	(4)
Quiz Solutions--Chapter 10	441	(3)
Quiz Solutions--Chapter 11	444	(5)
REFERENCES	449	(1)
INDEX	450