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An introduction to the theory of probability / Parimal Mukhopadhyay.

By: Mukhopadhyay, Parimal.
Material type: materialTypeLabelBookPublisher: Singapore : World Scientific Publishing Company, 2011Description: xvii, 474 p. ; 24 cm. + hbk.ISBN: 9789814313421; 9814313424.Subject(s): ProbabilitiesDDC classification: 519.2
Contents:
Preliminaries -- The classical approach -- Axiomatic approach -- Random variables and probability distributions -- Expectation of a discrete random variable -- Some properties of a probability distribution on R -- Generating functions -- Some discrete distributions on R1 -- Some continuous distributions on R1 -- Probability distribution on Rn -- Probability distributions of functions of random variables -- Convergence of a sequence of random variables -- Elements of stochastic process.
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 and convergence of random variables. Difficult ideas have been explained lucidly and have been augmented with explanatory notes, examples and exercises. The basic requirement for reading this book is simply a knowledge of mathematics at graduate level. This book tries to explain the difficult ideas in the axiomatic approach to the theory of probability in a clear and comprehensible manner. It includes several unusual distributions including the power series distribution that have been covered in great detail. and exercises with hints, which will make the book easily readable and engaging.The author is a former Professor of the Indian Statistical Institute, India.

Bibliography: (pages 461-463) and index.

Preliminaries -- The classical approach -- Axiomatic approach -- Random variables and probability distributions -- Expectation of a discrete random variable -- Some properties of a probability distribution on R -- Generating functions -- Some discrete distributions on R1 -- Some continuous distributions on R1 -- Probability distribution on Rn -- Probability distributions of functions of random variables -- Convergence of a sequence of random variables -- Elements of stochastic process.

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 and convergence of random variables. Difficult ideas have been explained lucidly and have been augmented with explanatory notes, examples and exercises. The basic requirement for reading this book is simply a knowledge of mathematics at graduate level. This book tries to explain the difficult ideas in the axiomatic approach to the theory of probability in a clear and comprehensible manner. It includes several unusual distributions including the power series distribution that have been covered in great detail. and exercises with hints, which will make the book easily readable and engaging.The author is a former Professor of the Indian Statistical Institute, India.

Table of contents provided by Syndetics

  • Preface (p. vii)
  • Glossary of Some Frequently Used Symbols and Abbreviations (p. 1)
  • 1 Preliminaries (p. 7)
  • 1.1 Introduction (p. 3)
  • 1.2 Historical Perspective (p. 4)
  • 1.3 Plan of the Book (p. 5)
  • 2 The Classical Approach (p. 61)
  • 2.1 Introduction (p. 9)
  • 2.2 Some Definitions (p. 9)
  • 2.2.1 Representation in set theory (p. 12)
  • 2.3 The Classical Definition of Probability (p. 15)
  • 2.4 Some Examples (p. 17)
  • 2.5 Models in Statistical Mechanics (p. 21)
  • 2.6 Some Theorems on Probability of Events (p. 23)
  • 2.6.1 Theorems on probability of union of events (p. 23)
  • 2.6.2 Conditional probability: Theorem of compound probability (p. 30)
  • 2.6.3 Independence of events (p. 33)
  • 2.6.4 Bayes theorem (p. 37)
  • 2.6.5 Binomial probability (p. 39)
  • 2.7 Limitations of Classical Definition (p. 40)
  • 2.8 Statistical or Empirical Definition (p. 41)
  • 2.9 Geometric Probability (p. 42)
  • 2.10 Further Examples (p. 46)
  • 2.11 Exercises and Complements (p. 53)
  • 3 Axiomatic Approach (p. 99)
  • 3.1 Introduction (p. 63)
  • 3.2 Set Algebra, Fields, ¿-Fields (p. 63)
  • 3.2.1 Algebra of sets (p. 63)
  • 3.2.2 Fields (p. 67)
  • 3.2.3 ¿-Field (p. 71)
  • 3.3 Point Function, Set Function (p. 76)
  • 3.3.1 Point function (p. 76)
  • 3.3.2 Set function (p. 76)
  • 3.3.3 Measure and measurable sets (p. 77)
  • 3.3.4 Inverse function (p. 79)
  • 3.4 Measurable Functions (p. 83)
  • 3.5 Axiomatic Definition of Probability (p. 87)
  • 3.5.1 Some simple properties (p. 89)
  • 3.6 Conditional Probability Measure (p. 92)
  • 3.7 Independent Trials and Product Space (p. 95)
  • 3.8 Exercises and Complements (p. 96)
  • 4 Random Variables and Probability Distributions (p. 124)
  • 4.1 Introduction (p. 101)
  • 4.2 Random Variables (p. 101)
  • 4.3 Induced Probability Space (p. 106)
  • 4.4 Probability Distribution Function of a Random Variable (p. 107)
  • 4.5 Discrete and Continuous Random Variables (p. 113)
  • 4.6 Independent Random Variables (p. 117)
  • 4.7 Integral of a Borel Measurable Function (Random Variable) (p. 118)
  • 4.7.1 The Lebesgue integral (p. 118)
  • 4.7.2 The Lebisgue-Stieltjes integral (p. 120)
  • 4.7.3 The Riemann-Stieltjes integral (p. 120)
  • 4.8 Exercises and Complements (p. 121)
  • 5 Expectation of a Discrete Random Variable (p. 138)
  • 5.1 Introduction (p. 125)
  • 5.2 Probability Distribution of a Discrete Random Variable (p. 125)
  • 5.3 Expectation (p. 126)
  • 5.4 Variance, Covariance, Correlation Coefficient (p. 130)
  • 5.5 Exercises and Complements (p. 137)
  • 6 Some Properties of a Probability Distribution on R (p. 186)
  • 6.1 Introduction (p. 139)
  • 6.2 Expectation (p. 139)
  • 6.2.1 Some properties of expectation (p. 142)
  • 6.3 Moments (p. 149)
  • 6.4 Some Moment Inequalities (p. 154)
  • 6.5 Moments of a Symmetric Probability Distribution (p. 160)
  • 6.6 Factorial Moments (p. 161)
  • 6.7 Different Measures of Central Tendency (p. 162)
  • 6.8 Measures of Dispersion (p. 164)
  • 6.9 Measures of Skewness and Kurtosis (p. 171)
  • 6.9.1 Measure of skewness (p. 171)
  • 6.9.2 Measure of kurtosis (p. 172)
  • 6.10 Some Probability Inequalities (p. 172)
  • 6.11 Exercises and Complements (p. 176)
  • Appendix 6.A (p. 185)
  • 7 Generating Functions (p. 214)
  • 7.1 Introduction (p. 187)
  • 7.2 Probability Generating Function (p. 187)
  • 7.3 Moment Generating Function (p. 192)
  • 7.4 Factorial Moment Generating Function (p. 197)
  • 7.5 Cumulant Generating Function (p. 197)
  • 7.6 Characteristic Function (p. 199)
  • 7.7 Exercises and Complements (p. 209)
  • 8 Some Discrete Distributions on R 1 (p. 247)
  • 8.1 Introduction (p. 215)
  • 8.2 The Discrete Uniform Distribution (p. 215)
  • 8.3 The Bernoulli Distribution (p. 216)
  • 8.4 The Binomial Distribution (p. 217)
  • 8.4.1 The truncated binomial distribution (p. 222)
  • 8.5 The Hypergeometric Distribution (p. 223)
  • 8.5.1 The positive hypergeometric distribution (p. 226)
  • 8.5.2 The negative hypergeometric distribution (p. 226)
  • 8.6 The Poisson Distribution (p. 226)
  • 8.6.1 The truncated Poisson distribution (p. 234)
  • 8.7 The Geometric Distribution (p. 234)
  • 8.8 The Negative Binomial Distribution (p. 236)
  • 8.8.1 The truncated negative binomial distribution (p. 239)
  • 8.9 The Power Series Distribution (p. 239)
  • 8.9.1 Some special cases (p. 241)
  • 8.10 Exercises and Complements (p. 244)
  • 9 Some Continuous Distributions on R 1 (p. 280)
  • 9.1 Introduction (p. 249)
  • 9.2 The Continuous Uniform Distribution (p. 249)
  • 9.3 The Exponential Distribution (p. 250)
  • 9.4 The Gamma Distribution and the Chi-Square Distribution (p. 252)
  • 9.4.1 The gamma function (p. 252)
  • 9.4.2 The gamma distribution (p. 254)
  • 9.4.3 The chi-square distribution (p. 255)
  • 9.5 The Beta Distribution (p. 257)
  • 9.5.1 The beta function (p. 257)
  • 9.5.2 The beta distribution (p. 257)
  • 9.6 The Cauchy Distribution (p. 260)
  • 9.7 The Normal Distribution (p. 262)
  • 9.7.1 The truncated normal distribution (p. 269)
  • 9.8 The Log-Normal Distribution (p. 270)
  • 9.9 The Double-Exponential (Laplace) Distribution (p. 273)
  • 9.10 The Pareto Distribution (p. 274)
  • 9.11 The Weibull Distribution (p. 275)
  • 9.12 The Extreme Value Distribution (p. 276)
  • 9.13 The Logistic Distribution (p. 276)
  • 9.14 Exercises and Complements (p. 276)
  • 10 Probability Distribution on R n (p. 328)
  • 10.1 Introduction (p. 281)
  • 10.2 Probability Distribution of a Random Vector (p. 281)
  • 10.3 Expectation and Moments of a Random Vector (p. 295)
  • 10.4 Multivariate Probability Generating Function (p. 300)
  • 10.5 Multivariate Moment Generating Function (p. 303)
  • 10.6 Multivariate Characteristic Function (p. 305)
  • 10.7 Conditional Expectation, Variance, Regression (p. 306)
  • 10.8 The Multinomial Distribution (p. 311)
  • 10.9 The Bivariate Normal Distribution (p. 314)
  • 10.10 Exercises and Complements (p. 319)
  • 11 Probability Distributions of Functions of Random Variables (p. 358)
  • 11.1 Introduction (p. 329)
  • 11.2 Functions of One Random Variable (p. 329)
  • 11.3 Probability Integral Transformation (p. 335)
  • 11.4 Functions of Two Random Variables (p. 337)
  • 11.4.1 Using distribution functions (p. 337)
  • 11.4.2 Transformation of variables (p. 340)
  • 11.5 Functions of n Random Variables (p. 344)
  • 11.6 Distributions of Maxima and Minima (p. 347)
  • 11.7 Use of Moment Generating Function (p. 350)
  • 11.8 Exercises and Complements (p. 351)
  • 12 Convergence of a Sequence of Random Variables (p. 406)
  • 12.1 Introduction (p. 359)
  • 12.2 Various Modes of Stochastic Convergence (p. 359)
  • 12.2.1 Convergence in probability (p. 360)
  • 12.2.2 Almost sure convergence (p. 362)
  • 12.2.3 Convergence in the rth mean (p. 365)
  • 12.2.4 Convergence in distribution (p. 369)
  • 12.2.5 Complete convergence (p. 377)
  • 12.2.6 Gradation of different modes of convergence (p. 378)
  • 12.3 Weak Law of Large Numbers (p. 379)
  • 12.3.1 Weierstrass approximation (p. 383)
  • 12.4 Strong Law of Large Numbers (p. 385)
  • 12.4.1 Kolmogorov's inequality and its ramifications (p. 391)
  • 12.4.2 Different SLLN's (p. 393)
  • 12.5 Central Limit Theorems (p. 395)
  • 12.6 Exercises and Complements (p. 402)
  • 13 Elements of Stochastic Process (p. 445)
  • 13.1 Introduction (p. 407)
  • 13.2 Preliminary Notions (p. 407)
  • 13.3 Markov Chain (p. 408)
  • 13.3.1 Classification of states (p. 412)
  • 13.3.2 Limits of higher order transition probabilities (p. 417)
  • 13.3.3 Irreducible chains (p. 419)
  • 13.3.4 Martingales (p. 423)
  • 13.4 Discrete Branching Process (p. 424)
  • 13.5 Simple Random Walk (p. 428)
  • 13.6 Continuous Time Markov Process (p. 431)
  • 13.6.1 Poisson process (p. 432)
  • 13.6.2 Birth process (p. 433)
  • 13.6.3 Birth and death process (p. 435)
  • 13.6.4 Normal process (p. 438)
  • 13.7 Exercises and Complements (p. 440)
  • Appendix (p. 459)
  • A.1 Random Variables as Limits (p. 447)
  • A.1.1 Approximating simple functions for independent random variables (p. 449)
  • A.2 Lebesgue Integration of a Borel Measurable Function (a Random Variable) (p. 449)
  • A.2.1 Integration of a random variable (p. 453)
  • A.2.2 The Lebesgue-Stieltjes integral (p. 456)
  • A.2.3 The Riemann-Stieltjes integral (p. 458)
  • Bibliography (p. 463)
  • Subject Index (p. 471)
  • Statistical Tables (p. 474)