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Note for Probability and Statistics - PS by Ashwin Kumar k

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The late MURRAY R. SPIEGEL received an MS degree in physics and a PhD in mathematics from Cornell University. He had positions at Harvard University, Columbia University, Oak Ridge, and Rensselaer Polytechnic Institute and served as a mathematical consultant at several large companies. His last position was professor and chairman of mathematics at Rensselaer Polytechnic Institute, Hartford Graduate Center. He was interested in most branches of mathematics, especially those which involve applications to physics and engineering problems. He was the author of numerous journal articles and 14 books on various topics in mathematics. JOHN J. SCHILLER is an associate professor of mathematics at Temple University. He received his PhD at the University of Pennsylvania. He has published research papers in the areas of Riemann surfaces, discrete mathematics, and mathematical biology. He has also coauthored texts in finite mathematics, precalculus, and calculus. R. ALU SRINIVASAN is a professor of mathematics at Temple University. He received his PhD at Wayne State University and has published extensively in probability and statistics. Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-179558-6 MHID: 0-07-179558-8 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-179557-9, MHID: 0-07-179557-X . All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at bulksales@mcgraw-hill.com. McGraw-Hill, the McGraw-Hill Publishing logo, Schaum’s, and related trade dress are trademarks or registered trademarks of The McGraw-Hill Companies and/or its affiliates in the United States and other countries and may not be used without written permission. All other trademarks are the property of their respective owners. The McGraw-Hill Companies is not associated with any product or vendor mentioned in this book. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.

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Preface to the Third Edition In the second edition of Probability and Statistics, which appeared in 2000, the guiding principle was to make changes in the first edition only where necessary to bring the work in line with the emphasis on topics in contemporary texts. In addition to refinements throughout the text, a chapter on nonparametric statistics was added to extend the applicability of the text without raising its level. This theme is continued in the third edition in which the book has been reformatted and a chapter on Bayesian methods has been added. In recent years, the Bayesian paradigm has come to enjoy increased popularity and impact in such areas as economics, environmental science, medicine, and finance. Since Bayesian statistical analysis is highly computational, it is gaining even wider acceptance with advances in computer technology. We feel that an introduction to the basic principles of Bayesian data analysis is therefore in order and is consistent with Professor Murray R. Spiegel’s main purpose in writing the original text—to present a modern introduction to probability and statistics using a background of calculus. J. SCHILLER R. A. SRINIVASAN Preface to the Second Edition The first edition of Schaum’s Probability and Statistics by Murray R. Spiegel appeared in 1975, and it has gone through 21 printings since then. Its close cousin, Schaum’s Statistics by the same author, was described as the clearest introduction to statistics in print by Gian-Carlo Rota in his book Indiscrete Thoughts. So it was with a degree of reverence and some caution that we undertook this revision. Our guiding principle was to make changes only where necessary to bring the text in line with the emphasis of topics in contemporary texts. The extensive treatment of sets, standard introductory material in texts of the 1960s and early 1970s, is considerably reduced. The definition of a continuous random variable is now the standard one, and more emphasis is placed on the cumulative distribution function since it is a more fundamental concept than the probability density function. Also, more emphasis is placed on the P values of hypotheses tests, since technology has made it possible to easily determine these values, which provide more specific information than whether or not tests meet a prespecified level of significance. Technology has also made it possible to eliminate logarithmic tables. A chapter on nonparametric statistics has been added to extend the applicability of the text without raising its level. Some problem sets have been trimmed, but mostly in cases that called for proofs of theorems for which no hints or help of any kind was given. Overall we believe that the main purpose of the first edition—to present a modern introduction to probability and statistics using a background of calculus—and the features that made the first edition such a great success have been preserved, and we hope that this edition can serve an even broader range of students. J. SCHILLER R. A. SRINIVASAN iii

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Preface to the First Edition The important and fascinating subject of probability began in the seventeenth century through efforts of such mathematicians as Fermat and Pascal to answer questions concerning games of chance. It was not until the twentieth century that a rigorous mathematical theory based on axioms, definitions, and theorems was developed. As time progressed, probability theory found its way into many applications, not only in engineering, science, and mathematics but in fields ranging from actuarial science, agriculture, and business to medicine and psychology. In many instances the applications themselves contributed to the further development of the theory. The subject of statistics originated much earlier than probability and dealt mainly with the collection, organization, and presentation of data in tables and charts. With the advent of probability it was realized that statistics could be used in drawing valid conclusions and making reasonable decisions on the basis of analysis of data, such as in sampling theory and prediction or forecasting. The purpose of this book is to present a modern introduction to probability and statistics using a background of calculus. For convenience the book is divided into two parts. The first deals with probability (and by itself can be used to provide an introduction to the subject), while the second deals with statistics. The book is designed to be used either as a textbook for a formal course in probability and statistics or as a comprehensive supplement to all current standard texts. It should also be of considerable value as a book of reference for research workers or to those interested in the field for self-study. The book can be used for a one-year course, or by a judicious choice of topics, a one-semester course. I am grateful to the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S., and to Longman Group Ltd., London, for permission to use Table III from their book Statistical Tables for Biological, Agricultural and Medical Research (6th edition, 1974). I also wish to take this opportunity to thank David Beckwith for his outstanding editing and Nicola Monti for his able artwork. M. R. SPIEGEL iv

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Contents Part I PROBABILITY 1 CHAPTER 1 Basic Probability 3 Random Experiments Sample Spaces Events The Concept of Probability The Axioms of Probability Some Important Theorems on Probability Assignment of Probabilities Conditional Probability Theorems on Conditional Probability Independent Events Bayes’ Theorem or Rule Combinatorial Analysis Fundamental Principle of Counting Tree Diagrams Permutations Combinations Binomial Coefficients Stirling’s Approximation to n! CHAPTER 2 Random Variables and Probability Distributions 34 Random Variables Discrete Probability Distributions Distribution Functions for Random Variables Distribution Functions for Discrete Random Variables Continuous Random Variables Graphical Interpretations Joint Distributions Independent Random Variables Change of Variables Probability Distributions of Functions of Random Variables Convolutions Conditional Distributions Applications to Geometric Probability CHAPTER 3 Mathematical Expectation 75 Definition of Mathematical Expectation Functions of Random Variables Some Theorems on Expectation The Variance and Standard Deviation Some Theorems on Variance Standardized Random Variables Moments Moment Generating Functions Some Theorems on Moment Generating Functions Characteristic Functions Variance for Joint Distributions. Covariance Correlation Coefficient Conditional Expectation, Variance, and Moments Chebyshev’s Inequality Law of Large Numbers Other Measures of Central Tendency Percentiles Other Measures of Dispersion Skewness and Kurtosis CHAPTER 4 Special Probability Distributions 108 The Binomial Distribution Some Properties of the Binomial Distribution The Law of Large Numbers for Bernoulli Trials The Normal Distribution Some Properties of the Normal Distribution Relation Between Binomial and Normal Distributions The Poisson Distribution Some Properties of the Poisson Distribution Relation Between the Binomial and Poisson Distributions Relation Between the Poisson and Normal Distributions The Central Limit Theorem The Multinomial Distribution The Hypergeometric Distribution The Uniform Distribution The Cauchy Distribution The Gamma Distribution The Beta Distribution The Chi-Square Distribution Student’s t Distribution The F Distribution Relationships Among Chi-Square, t, and F Distributions The Bivariate Normal Distribution Miscellaneous Distributions v

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