Title Handbook of probability / Ionut Florescu, Ciprian Tudor

Published Hoboken, New Jersey : Wiley, 2013

Table of Contents
	List of Figures					xv
	Preface					xvii
	Introduction					xix
1.	Probability Space					1
1.1.	Introduction/Purpose of the Chapter					1
1.2.	Vignette/Historical Notes					2
1.3.	Notations and Definitions					2
1.4.	Theory and Applications					4
1.4.1.	Algebras					4
1.4.2.	Sigma Algebras					5
1.4.3.	Measurable Spaces					7
1.4.4.	Examples					7
1.4.5.	The Borel σ-Algebra					9
1.5.	Summary					12
	Exercises					12
2.	Probability Measure					15
2.1.	Introduction/Purpose of the Chapter					15
2.2.	Vignette/Historical Notes					16
2.3.	Theory and Applications					17
2.3.1.	Definition and Basic Properties					17
2.3.2.	Uniqueness of Probability Measures					22
2.3.3.	Monotone Class					24
2.3.4.	Examples					26
2.3.5.	Monotone Convergence Properties of Probability					28
2.3.6.	Conditional Probability					31
2.3.7.	Independence of Events and σ-Fields					39
2.3.8.	Borel---Cantelli Lemmas					46
2.3.9.	Fatou's Lemmas					48
2.3.10.	Kolmogorov's Zero---One Law					49
2.4.	Lebesgue Measure on the Unit Interval (0,1					50
	Exercises					52
3.	Random Variables: Generalities					63
3.1.	Introduction/Purpose of the Chapter					63
3.2.	Vignette/Historical Notes					63
3.3.	Theory and Applications					64
3.3.1.	Definition					64
3.3.2.	The Distribution of a Random Variable					65
3.3.3.	The Cumulative Distribution Function of a Random Variable					67
3.3.4.	Independence of Random Variables					70
	Exercises					71
4.	Random Variables: The Discrete Case					79
4.1.	Introduction/Purpose of the Chapter					79
4.2.	Vignette/Historical Notes					80
4.3.	Theory and Applications					80
4.3.1.	Definition and Basic Facts					80
4.3.2.	Moments					84
4.4.	Examples of Discrete Random Variables					89
4.4.1.	The (Discrete) Uniform Distribution					89
4.4.2.	Bernoulli Distribution					91
4.4.3.	Binomial (n, p) Distribution					92
4.4.4.	Geometric (p) Distribution					95
4.4.5.	Negative Binomial (r, p) Distribution					101
4.4.6.	Hypergeometric Distribution (N, m, n)					102
4.4.7.	Poisson Distribution					104
	Exercises					108
5.	Random Variables: The continuous case					119
5.1.	Introduction/Purpose of the Chapter					119
5.2.	Vignette/Historical Notes					119
5.3.	Theory and Applications					120
5.3.1.	Probability Density Function (p.d.f.)					120
5.3.2.	Cumulative Distribution Function (c.d.f.)					124
5.3.3.	Moments					127
5.3.4.	Distribution of a Function of the Random Variable					128
5.4.	Examples					130
5.4.1.	Uniform Distribution on an Interval [a,b]					130
5.4.2.	Exponential Distribution					133
5.4.3.	Normal Distribution (μ, σ2)					136
5.4.4.	Gamma Distribution					139
5.4.5.	Beta Distribution					144
5.4.6.	Student's t Distribution					147
5.4.7.	Pareto Distribution					149
5.4.8.	The Log-Normal Distribution					151
5.4.9.	Laplace Distribution					153
5.4.10.	Double Exponential Distribution					155
	Exercises					156
6.	Generating Random Variables					177
6.1.	Introduction/Purpose of the Chapter					177
6.2.	Vignette/Historical Notes					178
6.3.	Theory and Applications					178
6.3.1.	Generating One-Dimensional Random Variables by Inverting the Cumulative Distribution Function (c.d.f.)					178
6.3.2.	Generating One-Dimensional Normal Random Variables					183
6.3.3.	Generating Random Variables. Rejection Sampling Method					186
6.3.4.	Generating from a Mixture of Distributions					193
6.3.5.	Generating Random Variables. Importance Sampling					195
6.3.6.	Applying Importance Sampling					198
6.3.7.	Practical Consideration: Normalizing Distributions					201
6.3.8.	Sampling Importance Resampling					203
6.3.9.	Adaptive Importance Sampling					204
6.4.	Generating Multivariate Distributions with Prescribed Covariance Structure					205
	Exercises					208
7.	Random Vectors in Rn					210
7.1.	Introduction/Purpose of the Chapter					210
7.2.	Vignette/Historical Notes					210
7.3.	Theory and Applications					211
7.3.1.	The Basics					211
7.3.2.	Marginal Distributions					212
7.3.3.	Discrete Random Vectors					214
7.3.4.	Multinomial Distribution					219
7.3.5.	Testing Whether Counts are Coming from a Specific Multinomial Distribution					220
7.3.6.	Independence					221
7.3.7.	Continuous Random Vectors					223
7.3.8.	Change of Variables. Obtaining Densities of Functions of Random Vectors					229
7.3.9.	Distribution of Sums of Random Variables. Convolutions					231
	Exercises					236
8.	Characteristic Function					255
8.1.	Introduction/Purpose of the Chapter					255
8.2.	Vignette/Historical Notes					255
8.3.	Theory and Applications					256
8.3.1.	Definition and Basic Properties					256
8.3.2.	The Relationship Between the Characteristic Function and the Distribution					260
8.4.	Calculation of the Characteristic Function for Commonly Encountered Distributions					265
8.4.1.	Bernoulli and Binomial					265
8.4.2.	Uniform Distribution					266
8.4.3.	Normal Distribution					267
8.4.4.	Poisson Distribution					267
8.4.5.	Gamma Distribution					268
8.4.6.	Cauchy Distribution					269
8.4.7.	Laplace Distribution					270
8.4.8.	Stable Distributions. Levy Distribution					271
8.4.9.	Truncated Levy Flight Distribution					274
	Exercises					275
9.	Moment-Generating Function					280
9.1.	Introduction/Purpose of the Chapter					280
9.2.	Vignette/Historical Notes					280
9.3.	Theory and Applications					281
9.3.1.	Generating Functions and Applications					281
9.3.2.	Moment-Generating Functions. Relation with the Characteristic Functions					288
9.3.3.	Relationship with the Characteristic Function					292
9.3.4.	Properties of the MGF					292
	Exercises					294
10.	Gaussian random vectors					300
10.1.	Introduction/Purpose of the Chapter					300
10.2.	Vignette/Historical Notes					301
10.3.	Theory and Applications					301
10.3.1.	The Basics					301
10.3.2.	Equivalent Definitions of a Gaussian Vector					303
10.3.3.	Uncorrelated Components and Independence					309
10.3.4.	The Density of a Gaussian Vector					313
10.3.5.	Cochran's Theorem					316
10.3.6.	Matrix Diagonalization and Gaussian Vectors					319
	Exercises					325
11.	Convergence Types. Almost Sure Convergence. LP-Convergence. Convergence in Probability					338
11.1.	Introduction/Purpose of the Chapter					338
11.2.	Vignette/Historical Notes					339
11.3.	Theory and Applications: Types of Convergence					339
11.3.1.	Traditional, Deterministic Convergence Types					339
11.3.2.	Convergence of Moments of an r.v.---Convergence in LP					341
11.3.3.	Almost Sure (a.s.) Convergence					342
11.3.4.	Convergence in Probability					344
11.4.	Relationships Between Types of Convergence					346
11.4.1.	a.s. and LP					347
11.4.2.	Probability and a.s.ILP?					351
11.4.3.	Uniform Integrability					357
	Exercises					359
12.	Limit Theorems					372
12.1.	Introduction/Purpose of the Chapter					372
12.2.	Vignette/Historical Notes					372
12.3.	Theory and Applications					375
12.3.1.	Weak Convergence					375
12.3.2.	The Law of Large Numbers					384
12.4.	Central Limit Theorem					401
	Exercises					409
13.	Appendix A: Integration Theory, general Expectations					421
13.1.	Integral of Measurable Functions					422
13.1.1.	Integral of Simple (Elementary) Functions					422
13.1.2.	Integral of Positive Measurable Functions					424
13.1.3.	Integral of Measurable Functions					428
13.2.	General Expectations and Moments of a Random Variable					429
13.2.1.	Moments and Central Moments. LP Space					430
13.2.2.	Variance and the Correlation Coefficient					431
13.2.3.	Convergence Theorems					433
14.	Appendix B: Inequalities Involving Random Variables and Their Expectations					434
14.1.	Functions of Random Variables. The Transport Formula					441
	Bibliography					445
	Index					447

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Description 1 online resource Contents Probability Space -- Probability Measure -- Random Variables : Generalities -- Random Variables : the Discrete Case -- Random Variables : the Continuous Case -- Generating Random Variables -- Random Vectors in Rn -- Characteristic Function -- Moment Generating Function -- Gaussian Random Vectors -- Convergence Types : A.S. Convergence, LP Convergence, Convergence in Probability -- Limit Theorems -- Appendix A: Integration Theory : General Expectations -- Appendix B: Inequalities Involving Random Variables and Their Expectations Summary THE COMPLETE COLLECTION NECESSARY FOR A CONCRETE UNDERSTANDING OF PROBABILITY Written in a clear, accessible, and comprehensive manner, the Handbook of Probability presents the fundamentals of probability with an emphasis on the balance of theory, application, and methodology. Utilizing basic examples throughout, the handbook expertly transitions between concepts and practice to allow readers an inclusive introduction to the field of probability. The book provides a useful format with self-contained chapters, allowing the reader easy and quick reference. Each chapter includes an introductio Notes "Published simultaneously in Canada"--Title page verso Bibliography Includes bibliographical references and index Notes Print version record and CIP data provided by publisher Subject Probabilities. probability. MATHEMATICS -- Applied. MATHEMATICS -- Probability & Statistics -- General. Probabilities Wahrscheinlichkeitsrechnung Form Electronic book Author Tudor, Ciprian, 1973- LC no. 2013020958 ISBN 9781118593141 1118593146 9781118593097 111859309X 9781118592960 1118592964 9781118593103 1118593103 0470647272 9780470647271

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