An Introduction to Probability and Statistics by Vijay K Rohatgi, A K MD Ehsanes

1. Probability 1 1.1 Introduction, 1 1.2 Sample Space, 2 1.3 Probability Axioms, 7 1.4 Combinatorics: Probability on Finite Sample Spaces, 21 1.5 Conditional Probability and Bayes Theorem, 28 1. 6 Independence of Events, ...

1. Probability 1 1.1 Introduction, 1 1.2 Sample Space, 2 1.3 Probability Axioms, 7 1.4 Combinatorics: Probability on Finite Sample Spaces, 21 1.5 Conditional Probability and Bayes Theorem, 28 1. 6 Independence of Events, 33 2. Random Variables and Their Probability Distributions 40 2.1 Introduction, 40 2.2 Random Variables, 40 2.3 Probability Distribution of a Random Variable, 43 2.4 Discrete and Continuous Random Variables, 48 2.5 Functions of a Random Variable, 57 3. Moments and Generating Functions 69 3.1 Introduction, 69 3.2 Moments of a Distribution Function, 69 3.3 Generating Functions, 85 3.4 Some Moment Inequalities, 95 4. Multiple Random Variables 102 4.1 Introduction, 102 vii CONTENTS 4.2 Multiple Random Variables, 102 4.3 Independent Random Variables, 119 4.4 Functions of Several Random Variables, 127 4.5 Covariance, Correlation, and Moments, 149 4.6 Conditional Expectation, 164 4.7 Order Statistics and Their Distributions, 171 Some Special Distributions 180 5.1 Introduction, 180 5.2 Some Discrete Distributions, 180 5.3 Some Continuous Distributions, 204 5.4 Bivariate and Multivariate Normal Distributions, 238 5.5 Exponential Family of Distributions, 251 Limit Theorems 256 6.1 Introduction, 256 6.2 Modes of Convergence, 256 6.3 Weak Law of Large Numbers, 274 6.4 Strong Law of Large Numbers, 281 6.5 Limiting Moment Generating Functions, 289 6.6 Central Limit Theorem, 293 Sample Moments and Their Distributions 306 7.1 Introduction, 306 7.2 Random Sampling, 307 7.3 Sample Characteristics and Their Distributions, 310 7.4 Chi-Square, f-, and F-Distributions: Exact Sampling Distributions, 324 7.5 Large-Sample Theory, 334 7.6 Distribution of (X, S2) in Sampling from a Normal Population, 339 7.7 Sampling from a Bivariate Normal Distribution, 344 Parametric Point Estimation 353 8.1 Introduction, 353 8.2 Problem of Point Estimation, 354 8.3 Sufficiency, Completeness, and Ancillarity, 358 8.4 Unbiased Estimation, 377 8.5 Unbiased Estimation (Continued): Lower Bound for the Variance of an Estimator, 391