Probability, Statistics, and Stochastic Processes

by Olofsson, Peter; Andersson, Mikael

• ISBN13: 9780470889749
• ISBN10: 0470889748
• Edition: 2nd
• Format: Hardcover
• Copyright: 2012-05-22
• Publisher: Wiley

Summary

This book provides a unique and balanced approach to probability, statistics, and stochastic processes. Readers gain a solid foundation in all three fields that serves as a stepping stone to more advanced investigations into each area. The Second Edition features new coverage of analysis of variance (ANOVA), consistency and efficiency of estimators, asymptotic theory for maximum likelihood estimators, empirical distribution function and the Kolmogorov-Smirnov test, general linear models, multiple comparisons, Markov chain Monte Carlo (MCMC), Brownian motion, martingales, and renewal theory. Many new introductory problems and exercises have also been added. This book combines a rigorous, calculus-based development of theory with a more intuitive approach that appeals to readers sense of reason and logic, an approach developed through the authors many years of classroom experience. The book begins with three chapters that develop probability theory and introduce the axioms of probability, random variables, and joint distributions. The next two chapters introduce limit theorems and simulation. Also included is a chapter on statistical inference with a focus on Bayesian statistics, which is an important, though often neglected, topic for undergraduate-level texts. Markov chains in discrete and continuous time are also discussed within the book. More than 400 examples are interspersed throughout to help illustrate concepts and theory and to assist readers in developing an intuitive sense of the subject. Readers will find many of the examples to be both entertaining and thought provoking. This is also true for the carefully selected problems that appear at the end of each chapter.

Author Biography

Peter Olofsson, PhD, is Professor in the Mathematics Department at Trinity University. Dr. Olofsson's research interests include stochastic processes, branching processes, mathematical biology, and Poisson approximation. He is the author of Probabilities: The Little Numbers That Rule Our Lives, also published by Wiley. Mikael Andersson, PhD, is Associate Professor in the Department of Applied Statistics at the Swedish University of Agricultural Sciences. Dr. Andersson's research interests include stochastic modeling of infectious diseases, epidemiology, and biological applications.

Table of Contents

Preface	p. xi
Preface to the First Edition	p. xiii
Basic Probability Theory	p. 1
Introduction	p. 1
Sample Spaces and Events	p. 3
The Axioms of Probability	p. 7
Finite Sample Spaces and Combinatorics	p. 15
Combinatorics	p. 17
Conditional Probability and Independence	p. 27
Independent Events	p. 33
The Law of Total Probability and Bayes' Formula	p. 41
Bayes' Formula	p. 47
Genetics and Probability	p. 54
Recursive Methods	p. 55
Problems	p. 63
Random Variables	p. 76
Introduction	p. 76
Discrete Random Variables	p. 77
Continuous Random Variables	p. 82
The Uniform Distribution	p. 90
Functions of Random Variables	p. 92
Expected Value and Variance	p. 95
The Expected Value of a Function of a Random Variable	p. 100
Variance of a Random Variable	p. 104
Special Discrete Distributions	p. 111
Indicators	p. 111
The Binomial Distribution	p. 112
The Geometric Distribution	p. 116
The Poisson Distribution	p. 117
The Hypergeometric Distribution	p. 121
Describing Data Sets	p. 121
The Exponential Distribution	p. 123
The Normal Distribution	p. 127
Other Distributions	p. 131
The Lognormal Distribution	p. 131
The Gamma Distribution	p. 133
The Cauchy Distribution	p. 134
Mixed Distributions	p. 135
Location Parameters	p. 137
The Failure Rate Function	p. 139
Uniqueness of the Failure Rate Function	p. 141
Problems	p. 144
Joint Distributions	p. 156
Introduction	p. 156
The Joint Distribution Function	p. 156
Discrete Random Vectors	p. 158
Jointly Continuous Random Vectors	p. 160
Conditional Distributions and Independence	p. 164
Independent Random Variables	p. 168
Functions of Random Vectors	p. 172
Real-Valued Functions of Random Vectors	p. 172
The Expected Value and Variance of a Sum	p. 176
Vector-Valued Functions of Random Vectors	p. 182
Conditional Expectation	p. 185
Conditional Expectation as a Random Variable	p. 189
Conditional Expectation and Prediction	p. 191
Conditional Variance	p. 192
Recursive Methods	p. 193
Covariance and Correlation	p. 196
The Correlation Coefficient	p. 201
The Bivariate Normal Distribution	p. 209
Multidimensional Random Vectors	p. 216
Order Statistics	p. 218
Reliability Theory	p. 223
The Multinomial Distribution	p. 225
The Multivariate Normal Distribution	p. 226
Convolution	p. 227
Generating Functions	p. 231
The Probability Generating Function	p. 231
The Moment Generating Function	p. 237
The Poisson Process	p. 240
Thinning and Superposition	p. 244
Problems	p. 247
Limit Theorems	p. 263
Introduction	p. 263
The Law of Large Numbers	p. 264
The Central Limit Theorem	p. 268
The Delta Method	p. 273
Convergence in Distribution	p. 275
Discrete Limits	p. 275
Continuous Limits	p. 277
Problems	p. 278
Simulation	p. 281
Introduction	p. 281
Random Number Generation	p. 282
Simulation of Discrete Distributions	p. 283
Simulation of Continuous Distributions	p. 285
Miscellaneous	p. 290
Problems	p. 292
Statistical Inference	p. 294
Introduction	p. 294
Point Estimators	p. 294
Estimating the Variance	p. 302
Confidence Intervals	p. 304
Confidence Interval for the Mean in the Normal Distribution with Known Variance	p. 307
Confidence Interval for an Unknown Probability	p. 308
One-Sided Confidence Intervals	p. 312
Estimation Methods	p. 312
The Method of Moments	p. 312
Maximum Likelihood	p. 315
Evaluation of Estimators with Simulation	p. 322
Bootstrap Simulation	p. 324
Hypothesis Testing	p. 327
Large Sample Tests	p. 332
Test for an Unknown Probability	p. 333
Further Topics in Hypothesis Testing	p. 334
P-Values	p. 334
Data Snooping	p. 335
The Power of a Test	p. 336
Multiple Hypothesis Testing	p. 338
Goodness of Fit	p. 339
Goodness-of-Fit Test for Independence	p. 346
Fisher's Exact Test	p. 349
Bayesian Statistics	p. 351
Noninformative priors	p. 359
Credibility Intervals	p. 362
Nonparametric Methods	p. 363
Nonparametric Hypothesis Testing	p. 363
Comparing Two Samples	p. 370
Nonparametric Confidence Intervals	p. 375
Problems	p. 378
Linear Models	p. 391
Introduction	p. 391
Sampling Distributions	p. 392
Single Sample Inference	p. 395
Inference for the Variance	p. 396
Inference for the Mean	p. 399
Comparing Two Samples	p. 402
Inference about Means	p. 402
Inference about Variances	p. 407
Analysis of Variance	p. 409
One-Way Analysis of Variance	p. 409
Multiple Comparisons: Tukey's Method	p. 412
Kruskal-Wallis Test	p. 413
Linear Regression	p. 415
Prediction	p. 422
Goodness of Fit	p. 424
The Sample Correlation Coefficient	p. 425
Spearman's Correlation Coefficient	p. 429
The General Linear Model	p. 431
Problems	p. 436
Stochastic Processes	p. 444
Introduction	p. 444
Discrete-Time Markov Chains	p. 445
Time Dynamics of a Markov Chain	p. 447
Classification of States	p. 450
Stationary Distributions	p. 454
Convergence to the Stationary Distribution	p. 460
Random Walks and Branching Processes	p. 464
The Simple Random Walk	p. 464
Multidimensional Random Walks	p. 468
Branching Processes	p. 469
Continuous-Time Markov Chains	p. 475
Stationary Distributions and Limit Distributions	p. 480
Birth-Death Processes	p. 484
Queueing Theory	p. 488
Further Properties of Queueing Systems	p. 491
Martingales	p. 494
Martingale Convergence	p. 495
Stopping Times	p. 497
Renewal Processes	p. 502
Asymptotic Properties	p. 504
Brownian Motion	p. 509
Hitting Times	p. 512
Variations of the Brownian Motion	p. 515
Problems	p. 517
Tables	p. 527
Answers to Selected Problems	p. 535
Further Reading	p. 551
Index	p. 553
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