Intuitive Probability And Random Processes Using Matlab

by Kay, Steven M.

• ISBN13: 9780387241579
• ISBN10: 0387241574
• Format: Hardcover
• Copyright: 2005-12-30
• Publisher: SPRINGER - VERLAG

Summary

This book covers:- The use of MATLAB examples to provide motivation for the theory to come.- The incorporation of MATLAB code to allow students to understand how the theory is applied in practice.- Numerous computer exercises to familiarize the student with MATLAB and how it is used to solve real problems.- The incorporation of "real-world" problems from various disciplines in each chapter to illlustrate the application of the chapter concepts.- Discussion of discrete random variables first, followed by continuous random variables to minimize confusion.

Author Biography

Steven M. Kay is a Professor of Electrical Engineering at the University of Rhode Island.

Table of Contents

Preface	vii
Introduction	1	(12)
What Is Probability?	1	(2)
Types of Probability Problems	3	(1)
Probabilistic Modeling	4	(3)
Analysis versus Computer Simulation	7	(1)
Some Notes to the Reader	8	(5)
References	9	(1)
Problems	10	(3)
Computer Simulation	13	(24)
Introduction	13	(1)
Summary	13	(1)
Why Use Computer Simulation?	14	(3)
Computer Simulation of Random Phenomena	17	(1)
Determining Characteristics of Random Variables	18	(6)
Real-World Example -- Digital Communications	24	(7)
References	26	(1)
Problems	26	(5)
Brief Introduction to MATLAB	31	(6)
Basic Probability	37	(36)
Introduction	37	(1)
Summary	37	(1)
Review of Set Theory	38	(5)
Assigning and Determining Probabilities	43	(5)
Properties of the Probability Function	48	(4)
Probabilities for Continuous Sample Spaces	52	(2)
Probabilities for Finite Sample Spaces -- Equally Likely Outcomes	54	(1)
Combinatorics	55	(7)
Binomial Probability Law	62	(2)
Real-World Example -- Quality Control	64	(9)
References	66	(1)
Problems	66	(7)
Conditional Probability	73	(32)
Introduction	73	(1)
Summary	73	(1)
Joint Events and the Conditional Probability	74	(9)
Statistically Independent Events	83	(3)
Bayes' Theorem	86	(3)
Multiple Experiments	89	(8)
Real-World Example -- Cluster Recognition	97	(8)
References	100	(1)
Problems	100	(5)
Discrete Random Variables	105	(28)
Introduction	105	(1)
Summary	105	(1)
Definition of Discrete Random Variable	106	(2)
Probability of Discrete Random Variables	108	(3)
Important Probability Mass Functions	111	(2)
Approximation of Binomial PMF by Poisson PMF	113	(2)
Transformation of Discrete Random Variables	115	(2)
Cumulative Distribution Function	117	(5)
Computer Simulation	122	(2)
Real-World Example -- Servicing Customers	124	(9)
References	128	(1)
Problems	128	(5)
Expected Values for Discrete Random Variables	133	(34)
Introduction	133	(1)
Summary	133	(1)
Determining Averages from the PMF	134	(3)
Expected Values of Some Important Random Variables	137	(3)
Expected Value for a Function of a Random Variable	140	(3)
Variance and Moments of a Random Variable	143	(4)
Characteristic Functions	147	(6)
Estimating Means and Variances	153	(2)
Real-World Example -- Data Compression	155	(8)
References	157	(1)
Problems	158	(5)
Derivation of E[g(X)] Formula	163	(2)
MATLAB Code Used to Estimate Mean and Variance	165	(2)
Multiple Discrete Random Variables	167	(48)
Introduction	167	(1)
Summary	168	(1)
Jointly Distributed Random Variables	169	(5)
Marginal PMFs and CDFs	174	(4)
Independence of Multiple Random Variables	178	(3)
Transformations of Multiple Random Variables	181	(5)
Expected Values	186	(3)
Joint Moments	189	(3)
Prediction of a Random Variable Outcome	192	(6)
Joint Characteristic Functions	198	(2)
Computer Simulation of Random Vectors	200	(2)
Real-World Example Assessing Health Risks	202	(11)
References	204	(1)
Problems	204	(9)
Derivation of the Cauchy-Schwarz Inequality	213	(2)
Conditional Probability Mass Functions	215	(32)
Introduction	215	(1)
Summary	216	(1)
Conditional Probability Mass Function	217	(3)
Joint, Conditional, and Marginal PMFs	220	(5)
Simplifying Probability Calculations using Conditioning	225	(4)
Mean of the Conditional PMF	229	(6)
Computer Simulation Based on Conditioning	235	(2)
Real-World Example -- Modeling Human Learning	237	(10)
References	240	(1)
Problems	240	(7)
Discrete N-Dimensional Random Variables	247	(38)
Introduction	247	(1)
Summary	247	(1)
Random Vectors and Probability Mass Functions	248	(3)
Transformations	251	(4)
Expected Values	255	(10)
Joint Moments and the Characteristic Function	265	(1)
Conditional Probability Mass Functions	266	(3)
Computer Simulation of Random Vectors	269	(3)
Real-World Example -- Image Coding	272	(13)
References	277	(1)
Problems	277	(8)
Continuous Random Variables	285	(58)
Introduction	285	(1)
Summary	286	(1)
Definition of a Continuous Random Variable	287	(6)
The PDF and Its Properties	293	(2)
Important PDFs	295	(8)
Cumulative Distribution Functions	303	(8)
Transformations	311	(6)
Mixed Random Variables	317	(7)
Computer Simulation	324	(4)
Real-World Example -- Setting Clipping Levels	328	(11)
References	331	(1)
Problems	331	(8)
Derivation of PDF of a Transformed Continuous Random Variable	339	(2)
MATLAB Subprograms to Compute Q and Inverse Q Functions	341	(2)
Expected Values for Continuous Random Variables	343	(34)
Introduction	343	(1)
Summary	343	(1)
Determining the Expected Value	344	(5)
Expected Values for Important PDFs	349	(2)
Expected Value for a Function of a Random Variable	351	(4)
Variance and Moments	355	(4)
Characteristic Functions	359	(2)
Probability, Moments, and the Chebyshev Inequality	361	(2)
Estimating the Mean and Variance	363	(1)
Real-World Example -- Critical Software Testing	364	(11)
References	367	(1)
Problems	367	(8)
Partial Proof of Expected Value of Function of Continuous Random Variable	375	(2)
Multiple Continuous Random Variables	377	(56)
Introduction	377	(1)
Summary	378	(1)
Jointly Distributed Random Variables	379	(8)
Marginal PDFs and the Joint CDF	387	(5)
Independence of Multiple Random Variables	392	(2)
Transformations	394	(10)
Expected Values	404	(8)
Joint Moments	412	(1)
Prediction of Random Variable Outcome	412	(2)
Joint Characteristic Functions	414	(1)
Computer Simulation	415	(4)
Real-World Example - Optical Character Recognition	419	(14)
References	423	(1)
Problems	423	(10)
Conditional Probability Density Functions	433	(24)
Introduction	433	(1)
Summary	433	(1)
Conditional PDF	434	(6)
Joint, Conditional, and Marginal PDFs	440	(4)
Simplifying Probability Calculations Using Conditioning	444	(2)
Mean of Conditional PDF	446	(1)
Computer Simulation of Jointly Continuous Random Variables	447	(2)
Real-World Example -- Retirement Planning	449	(8)
References	452	(1)
Problems	452	(5)
Continuous N-Dimensional Random Variables	457	(28)
Introduction	457	(1)
Summary	457	(1)
Random Vectors and PDFs	458	(5)
Transformations	463	(2)
Expected Values	465	(2)
Joint Moments and the Characteristic Function	467	(4)
Conditional PDFs	471	(1)
Prediction of a Random Variable Outcome	471	(4)
Computer Simulation of Gaussian Random Vectors	475	(1)
Real-World Example -- Signal Detection	476	(9)
References	479	(1)
Problems	479	(6)
Probability and Moment Approximations Using Limit Theorems	485	(30)
Introduction	485	(1)
Summary	486	(1)
Convergence and Approximation of a Sum	486	(1)
Law of Large Numbers	487	(5)
Central Limit Theorem	492	(11)
Real-World Example -- Opinion Polling	503	(8)
References	506	(1)
Problems	507	(4)
MATLAB Program to Compute Repeated Convolution of PDFs	511	(2)
Proof of Central Limit Theorem	513	(2)
Basic Random Processes	515	(32)
Introduction	515	(1)
Summary	516	(1)
What Is a Random Process?	517	(3)
Types of Random Processes	520	(3)
The Important, Property of Stationarity	523	(5)
Some More Examples	528	(5)
Joint Moments	533	(5)
Real-World Example - Statistical Data Analysis	538	(9)
References	542	(1)
Problems	542	(5)
Wide Sense Stationary Random Processes	547	(50)
Introduction	547	(1)
Summary	548	(1)
Definition of WSS Random Process	549	(3)
Autocorrelation Sequence	552	(10)
Ergodicity and Temporal Averages	562	(5)
The Power Spectral Density	567	(9)
Estimation of the ACS and PSD	576	(4)
Continuous-Time WSS Random Processes	580	(6)
Real-World Example - Random Vibration Testing	586	(11)
References	589	(1)
Problems	590	(7)
Linear Systems and Wide Sense Stationary Random Processes	597	(44)
Introduction	597	(1)
Summary	598	(1)
Random Process at Output of Linear System	598	(9)
Interpretation of the PSD	607	(2)
Wiener Filtering	609	(14)
Continuous-Time Definitions and Formulas	623	(3)
Real-World Example -- Speech Synthesis	626	(11)
References	630	(1)
Problems	631	(6)
Solution for Infinite Length Predictor	637	(4)
Multiple Wide Sense Stationary Random Processes	641	(32)
Introduction	641	(1)
Summary	642	(1)
Jointly Distributed WSS Random Processes	642	(5)
The Cross-Power Spectral Density	647	(5)
Transformations of Multiple Random Processes	652	(5)
Continuous-Time Definitions and Formulas	657	(4)
Cross-Correlation Sequence Estimation	661	(2)
Real-World Example -- Brain Physiology Research	663	(10)
References	667	(1)
Problems	667	(6)
Gaussian Random Processes	673	(38)
Introduction	673	(2)
Summary	675	(1)
Definition of the Gaussian Random Process	676	(5)
Linear Transformations	681	(2)
Nonlinear Transformations	683	(3)
Continuous-Time Definitions and Formulas	686	(3)
Special Continuous-Time Gaussian Random Processes	689	(7)
Computer Simulation	696	(2)
Real-World Example -- Estimating Fish Populations	698	(11)
References	701	(1)
Problems	702	(7)
MATLAB Listing for Figure 20.2	709	(2)
Poisson Random Processes	711	(28)
Introduction	711	(2)
Summary	713	(1)
Derivation of Poisson Counting Random Process	714	(4)
Interarrival Times	718	(3)
Arrival Times	721	(2)
Compound Poisson Random Process	723	(4)
Computer Simulation	727	(1)
Real-World Example -- Automobile Traffic Signal Planning	728	(9)
References	732	(1)
Problems	732	(5)
Joint PDF for Interarrival Times	737	(2)
Markov Chains	739	(38)
Introduction	739	(5)
Summary	744	(1)
Definitions	744	(4)
Computation of State Probabilities	748	(8)
Ergodic Markov Chains	756	(3)
Further Steady-State Characteristics	759	(3)
K-State Markov Chains	762	(2)
Computer Simulation	764	(1)
Real-World Example -- Strange Markov Chain Dynamics	765	(10)
References	767	(1)
Problems	767	(8)
Solving for the Stationary PMF	775	(2)
A. Glossary of Symbols and Abbrevations	777	(6)
B. Assorted Math Facts and Formulas	783	(6)
B.1 Proof by Induction	783	(1)
B.2 Trigonometry	784	(1)
B.3 Limits	784	(1)
B.4 Sums	785	(1)
B.5 Calculus	786	(3)
C. Linear and Matrix Algebra	789	(6)
C.1 Definitions	789	(2)
C.2 Special Matrices	791	(1)
C.3 Matrix Manipulation and Formulas	792	(1)
C.4 Some Properties of PD (PSD) Matrices	793	(1)
C.5 Eigendecomposition of Matrices	793	(2)
D. Summary of Signals, Linear Transforms, and Linear Systems	795	(14)
D.1 Discrete-Time Signals	795	(1)
D.2 Linear Transforms	796	(4)
D.3 Discrete-Time Linear Systems	800	(4)
D.4 Continuous-Time Signals	804	(1)
D.5 Linear Transforms	805	(2)
D.6 Continuous-Time Linear Systems	807	(2)
E. Answers to Selected Problems	809	(14)
Index	823