An Introduction to the Theory of Probability

by Mukhopadhyay, Parimal

• ISBN13: 9789814313421
• ISBN10: 9814313424
• Format: Hardcover
• Copyright: 2011-08-30
• Publisher: World Scientific Pub Co Inc

Summary

The Theory of Probability is a major tool that can be used to explain and understand the various phenomena in different natural, physical and social sciences. This book provides a systematic exposition of the theory in a setting which contains a balanced mixture of the classical approach and the modern day axiomatic approach. After reviewing the basis of the theory, the book considers univariate distributions, bivariate normal distribution, multinomial distribution, convergence of random variables and elements of stochastic process. Difficult ideas have been explained lucidly and augmented with explanatory notes, examples and exercises. The basic requirement for reading the book is the knowledge of mathematics at graduate level.

Table of Contents

Preface	p. vii
Glossary of Some Frequently Used Symbols and Abbreviations	p. 1
Preliminaries	p. 7
Introduction	p. 3
Historical Perspective	p. 4
Plan of the Book	p. 5
The Classical Approach	p. 61
Introduction	p. 9
Some Definitions	p. 9
Representation in set theory	p. 12
The Classical Definition of Probability	p. 15
Some Examples	p. 17
Models in Statistical Mechanics	p. 21
Some Theorems on Probability of Events	p. 23
Theorems on probability of union of events	p. 23
Conditional probability: Theorem of compound probability	p. 30
Independence of events	p. 33
Bayes theorem	p. 37
Binomial probability	p. 39
Limitations of Classical Definition	p. 40
Statistical or Empirical Definition	p. 41
Geometric Probability	p. 42
Further Examples	p. 46
Exercises and Complements	p. 53
Axiomatic Approach	p. 99
Introduction	p. 63
Set Algebra, Fields, ¿-Fields	p. 63
Algebra of sets	p. 63
Fields	p. 67
¿-Field	p. 71
Point Function, Set Function	p. 76
Point function	p. 76
Set function	p. 76
Measure and measurable sets	p. 77
Inverse function	p. 79
Measurable Functions	p. 83
Axiomatic Definition of Probability	p. 87
Some simple properties	p. 89
Conditional Probability Measure	p. 92
Independent Trials and Product Space	p. 95
Exercises and Complements	p. 96
Random Variables and Probability Distributions	p. 124
Introduction	p. 101
Random Variables	p. 101
Induced Probability Space	p. 106
Probability Distribution Function of a Random Variable	p. 107
Discrete and Continuous Random Variables	p. 113
Independent Random Variables	p. 117
Integral of a Borel Measurable Function (Random Variable)	p. 118
The Lebesgue integral	p. 118
The Lebisgue-Stieltjes integral	p. 120
The Riemann-Stieltjes integral	p. 120
Exercises and Complements	p. 121
Expectation of a Discrete Random Variable	p. 138
Introduction	p. 125
Probability Distribution of a Discrete Random Variable	p. 125
Expectation	p. 126
Variance, Covariance, Correlation Coefficient	p. 130
Exercises and Complements	p. 137
Some Properties of a Probability Distribution on R	p. 186
Introduction	p. 139
Expectation	p. 139
Some properties of expectation	p. 142
Moments	p. 149
Some Moment Inequalities	p. 154
Moments of a Symmetric Probability Distribution	p. 160
Factorial Moments	p. 161
Different Measures of Central Tendency	p. 162
Measures of Dispersion	p. 164
Measures of Skewness and Kurtosis	p. 171
Measure of skewness	p. 171
Measure of kurtosis	p. 172
Some Probability Inequalities	p. 172
Exercises and Complements	p. 176
Appendix 6.A	p. 185
Generating Functions	p. 214
Introduction	p. 187
Probability Generating Function	p. 187
Moment Generating Function	p. 192
Factorial Moment Generating Function	p. 197
Cumulant Generating Function	p. 197
Characteristic Function	p. 199
Exercises and Complements	p. 209
Some Discrete Distributions on R1	p. 247
Introduction	p. 215
The Discrete Uniform Distribution	p. 215
The Bernoulli Distribution	p. 216
The Binomial Distribution	p. 217
The truncated binomial distribution	p. 222
The Hypergeometric Distribution	p. 223
The positive hypergeometric distribution	p. 226
The negative hypergeometric distribution	p. 226
The Poisson Distribution	p. 226
The truncated Poisson distribution	p. 234
The Geometric Distribution	p. 234
The Negative Binomial Distribution	p. 236
The truncated negative binomial distribution	p. 239
The Power Series Distribution	p. 239
Some special cases	p. 241
Exercises and Complements	p. 244
Some Continuous Distributions on R1	p. 280
Introduction	p. 249
The Continuous Uniform Distribution	p. 249
The Exponential Distribution	p. 250
The Gamma Distribution and the Chi-Square Distribution	p. 252
The gamma function	p. 252
The gamma distribution	p. 254
The chi-square distribution	p. 255
The Beta Distribution	p. 257
The beta function	p. 257
The beta distribution	p. 257
The Cauchy Distribution	p. 260
The Normal Distribution	p. 262
The truncated normal distribution	p. 269
The Log-Normal Distribution	p. 270
The Double-Exponential (Laplace) Distribution	p. 273
The Pareto Distribution	p. 274
The Weibull Distribution	p. 275
The Extreme Value Distribution	p. 276
The Logistic Distribution	p. 276
Exercises and Complements	p. 276
Probability Distribution on Rn	p. 328
Introduction	p. 281
Probability Distribution of a Random Vector	p. 281
Expectation and Moments of a Random Vector	p. 295
Multivariate Probability Generating Function	p. 300
Multivariate Moment Generating Function	p. 303
Multivariate Characteristic Function	p. 305
Conditional Expectation, Variance, Regression	p. 306
The Multinomial Distribution	p. 311
The Bivariate Normal Distribution	p. 314
Exercises and Complements	p. 319
Probability Distributions of Functions of Random Variables	p. 358
Introduction	p. 329
Functions of One Random Variable	p. 329
Probability Integral Transformation	p. 335
Functions of Two Random Variables	p. 337
Using distribution functions	p. 337
Transformation of variables	p. 340
Functions of n Random Variables	p. 344
Distributions of Maxima and Minima	p. 347
Use of Moment Generating Function	p. 350
Exercises and Complements	p. 351
Convergence of a Sequence of Random Variables	p. 406
Introduction	p. 359
Various Modes of Stochastic Convergence	p. 359
Convergence in probability	p. 360
Almost sure convergence	p. 362
Convergence in the rth mean	p. 365
Convergence in distribution	p. 369
Complete convergence	p. 377
Gradation of different modes of convergence	p. 378
Weak Law of Large Numbers	p. 379
Weierstrass approximation	p. 383
Strong Law of Large Numbers	p. 385
Kolmogorov's inequality and its ramifications	p. 391
Different SLLN's	p. 393
Central Limit Theorems	p. 395
Exercises and Complements	p. 402
Elements of Stochastic Process	p. 445
Introduction	p. 407
Preliminary Notions	p. 407
Markov Chain	p. 408
Classification of states	p. 412
Limits of higher order transition probabilities	p. 417
Irreducible chains	p. 419
Martingales	p. 423
Discrete Branching Process	p. 424
Simple Random Walk	p. 428
Continuous Time Markov Process	p. 431
Poisson process	p. 432
Birth process	p. 433
Birth and death process	p. 435
Normal process	p. 438
Exercises and Complements	p. 440
Appendix	p. 459
Random Variables as Limits	p. 447
Approximating simple functions for independent random variables	p. 449
Lebesgue Integration of a Borel Measurable Function (a Random Variable)	p. 449
Integration of a random variable	p. 453
The Lebesgue-Stieltjes integral	p. 456
The Riemann-Stieltjes integral	p. 458
Bibliography	p. 463
Subject Index	p. 471
Statistical Tables	p. 474
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