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Description:

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… More 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 author′s 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.Less

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Book details

Edition: 2nd Copyright year: 2012 Publisher: John Wiley & Sons, Limited Publication date: 6/8/2012 Binding: Hardcover Pages: 576 Size: 6.25" wide x 9.25" long x 1.25" tall Weight: 2.310 Language: English

AuthorTable of Contents

Preface

Preface to the First Edition

Basic Probability Theory

Introduction

Sample Spaces and Events

The Axioms of Probability

Finite Sample Spaces and Combinatorics

Combinatorics

Conditional Probability and Independence

Independent Events

The Law of Total Probability and Bayes' Formula

Bayes' Formula

Genetics and Probability

Recursive Methods

Problems

Random Variables

Introduction

Discrete Random Variables

Continuous Random Variables

The Uniform Distribution

Functions of Random Variables

Expected Value and Variance

The Expected Value of a Function of a Random Variable

Variance of a Random Variable

Special Discrete Distributions

Indicators

The Binomial Distribution

The Geometric Distribution

The Poisson Distribution

The Hypergeometric Distribution

Describing Data Sets

The Exponential Distribution

The Normal Distribution

Other Distributions

The Lognormal Distribution

The Gamma Distribution

The Cauchy Distribution

Mixed Distributions

Location Parameters

The Failure Rate Function

Uniqueness of the Failure Rate Function

Problems

Joint Distributions

Introduction

The Joint Distribution Function

Discrete Random Vectors

Jointly Continuous Random Vectors

Conditional Distributions and Independence

Independent Random Variables

Functions of Random Vectors

Real-Valued Functions of Random Vectors

The Expected Value and Variance of a Sum

Vector-Valued Functions of Random Vectors

Conditional Expectation

Conditional Expectation as a Random Variable

Conditional Expectation and Prediction

Conditional Variance

Recursive Methods

Covariance and Correlation

The Correlation Coefficient

The Bivariate Normal Distribution

Multidimensional Random Vectors

Order Statistics

Reliability Theory

The Multinomial Distribution

The Multivariate Normal Distribution

Convolution

Generating Functions

The Probability Generating Function

The Moment Generating Function

The Poisson Process

Thinning and Superposition

Problems

Limit Theorems

Introduction

The Law of Large Numbers

The Central Limit Theorem

The Delta Method

Convergence in Distribution

Discrete Limits

Continuous Limits

Problems

Simulation

Introduction

Random Number Generation

Simulation of Discrete Distributions

Simulation of Continuous Distributions

Miscellaneous

Problems

Statistical Inference

Introduction

Point Estimators

Estimating the Variance

Confidence Intervals

Confidence Interval for the Mean in the Normal Distribution with Known Variance

Confidence Interval for an Unknown Probability

One-Sided Confidence Intervals

Estimation Methods

The Method of Moments

Maximum Likelihood

Evaluation of Estimators with Simulation

Bootstrap Simulation

Hypothesis Testing

Large Sample Tests

Test for an Unknown Probability

Further Topics in Hypothesis Testing

P-Values

Data Snooping

The Power of a Test

Multiple Hypothesis Testing

Goodness of Fit

Goodness-of-Fit Test for Independence

Fisher's Exact Test

Bayesian Statistics

Noninformative priors

Credibility Intervals

Nonparametric Methods

Nonparametric Hypothesis Testing

Comparing Two Samples

Nonparametric Confidence Intervals

Problems

Linear Models

Introduction

Sampling Distributions

Single Sample Inference

Inference for the Variance

Inference for the Mean

Comparing Two Samples

Inference about Means

Inference about Variances

Analysis of Variance

One-Way Analysis of Variance

Multiple Comparisons: Tukey's Method

Kruskal-Wallis Test

Linear Regression

Prediction

Goodness of Fit

The Sample Correlation Coefficient

Spearman's Correlation Coefficient

The General Linear Model

Problems

Stochastic Processes

Introduction

Discrete-Time Markov Chains

Time Dynamics of a Markov Chain

Classification of States

Stationary Distributions

Convergence to the Stationary Distribution

Random Walks and Branching Processes

The Simple Random Walk

Multidimensional Random Walks

Branching Processes

Continuous-Time Markov Chains

Stationary Distributions and Limit Distributions

Birth-Death Processes

Queueing Theory

Further Properties of Queueing Systems

Martingales

Martingale Convergence

Stopping Times

Renewal Processes

Asymptotic Properties

Brownian Motion

Hitting Times

Variations of the Brownian Motion

Problems

Tables

Answers to Selected Problems

Further Reading

Index

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