Probability Theory with Applications

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The material in this book is designed for a standard graduate course on probability theory, including some important applications. It was prepared from the sets of lecture notes for a course that I have taught several times over the past 20 years. The present version reflects the reactions of my audiences as well as some of the textbooks that I used.

Audience: Standard Graduate course on probability Theory.

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Editorial Reviews

From The Critics
Reviewer: Michael J. Schrift, DO(University of Illinois at Chicago College of Medicine)
Description: One of the ethical principles, beneficence, as described in the Belmont Report and codified in The Common Rule ( 45 C.F.R.Part 46), is an obligation to do no harm and to "maximize possible benefits and minimize possible harms" to the individual research participant. In research this implies that "one should not injure one person regardless of the benefits that might come to others." At times, though, one cannot know that something is harmful until it is tried and in the process of trying, or experimentation, persons may be exposed to risk of harm. Investigators and their institutions have to plan to maximize benefits and minimize risks and since it is not possible to remove all risks from the research process, the research needs to have continuing review and monitoring of data to ensure that the study does not continue after the emergence of reliable evidence of reduced efficacy and/or safety. That's where this essential book comes in: what are the statistics and underlying mathematics of the so-called "stopping rules" that data safety monitoring committees employ? Written by a physician-scientist who also happens to be a biostatistician, this book should be required reading for all principal investigators, data safety monitoring committee members, and research ethics committee members.
Purpose: The author correctly notes that clinical investigators with little quantitative background frequently have difficulty communicating with biostatisticians and research methodologists. This book is intended to bring clinical investigators up to speed and toremedy their statistical/mathematical challenges. As he notes, "If you know nothing about monitoring guidelines in clinical trials, then this book is for you."
Audience: The intended audience includes clinical investigators and data monitoring committee members.
Features: The first of the book's eight chapters is an overview of the history of monitoring procedures, understanding randomized blinded controlled trials, and the ethical concerns of this methodology. Chapter 2 covers basic statistics in research including the principles of probability. Chapter 3 summarizes the various monitoring procedures that are used in clinical trials. Chapters 4 and 5 focus on path analysis and group sequential analysis respectively. Chapter 6 helps the reader understand the circumstances in which a clinical trial may be stopped early for a beneficial finding based on a conditional power approach. The monitoring procedures to identify the harmful effects of the tested intervention are detailed in Chapter 7. Lastly, Bayesian statistics are introduced in chapter 8 in the use of monitoring procedures. The appendixes were difficult for me to understand mathematically, but for those so inclined, they include such topics as boundary values for normal mean, conditional Brownian motion, boundary values and conditional power, supporting Bayesian computations, and standard normal probabilities. The index was useful.
Assessment: Written by a clinical researcher/physician/biostatistician, this book is an excellent way to learn the statistics and mathematics underlying data safety monitoring. I highly recommend it.
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Product Details

Table of Contents

Preface. List of Symbols. Background Material and Preliminaries. Independence and Strong Convergence. Conditioning and Some Dependence Classes. Probability Distributions and Characteristic Functions. Weak Limit Laws. Application to Some Statistical Estimation Problems. Stopping Times, Martingales, and Convergence. Limit Laws for Some Dependent Sequences. A Glimpse into Stochastic Processes. References. Author Index. Subject Index.

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