Hwei P. Hsu - Schaums Outline of Probability, Random Variables, and Random Processes, 4th Edition

HWEI P. HSU received his BS from National Taiwan University and his MS and PhD from Case Institute of Technology. He has published several books which include Schaums Outline of Analog and Digital Communications and Schaums Outline of Signals and Systems. Copyright 2020, 2013, 2009, 1998 by McGraw-Hill Education. 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: 978-1-26-045382-9
MHID: 1-26-045382-0 The material in this eBook also appears in the print version of this title: ISBN: 978-1-26-045381-2, MHID: 1-26-045381-2. eBook conversion by codeMantra
Version 1.0 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 Education 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 visit the Contact Us page at www.mhprofessional.com.

McGraw-Hill Education, the McGraw-Hill Education logo, Schaums, and related trade dress are trademarks or registered trademarks of McGraw-Hill Education 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. McGraw-Hill Education is not associated with any product or vendor mentioned in this book. TERMS OF USE This is a copyrighted work and McGraw-Hill Education 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 Educations 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 EDUCATION 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 Education 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 Education 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 Education has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill Education 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.

The purpose of this book, like its previous edition, is to provide an introduction to the principles of probability, random variables, and random processes and their applications. The book is designed for students in various disciplines of engineering, science, mathematics, and management. The background required to study the book is one year of calculus, elementary differential equations, matrix analysis, and some signal and system theory, including Fourier transforms.

The book can be used as a self-contained textbook or for self-study. Each topic is introduced in a chapter with numerous solved problems. The solved problems constitute an integral part of the text. This new edition includes and expands the contents of the first edition. In addition to refinement through the text, two new sections on probability-generating functions and martingales have been added and a new chapter on information theory has been added. I wish to thank my granddaughter Elysia Ann Krebs for helping me in the preparation of this revision.

I also wish to express my appreciation to the editorial staff of the McGraw-Hill Schaums Series for their care, cooperation, and attention devoted to the preparation of the book. HWEI P. HSU
Shannondell at Valley Forge, Audubon, Pennsylvania

The purpose of this book is to provide an introduction to the principles of probability, random variables, and random processes and their applications. The book is designed for students in various disciplines of engineering, science, mathematics, and management. It may be used as a textbook and/or a supplement to all current comparable texts. It should also be useful to those interested in the field of self-study.

The book combines the advantages of both the textbook and the so-called review book. It provides the textual explanations of the textbook, and in the direct way characteristic of the review book, it gives hundreds of completely solved problems that use essential theory and techniques. Moreover, the solved problems are an integral part of the text. The background required to study the book is one year of calculus, elementary differential equations, matrix analysis, and some signal and system theory, including Fourier transforms. I wish to thank Dr. Gordon Silverman for his invaluable suggestions and critical review of the manuscript.

I also wish to express my appreciation to the editorial staff of the McGraw-Hill Schaum Series for their care, cooperation, and attention devoted to the preparation of the book. Finally, I thank my wife, Daisy, for her patience and encouragement. HWEI P. HSU
Montville, New Jersey

*The laptop icon next to an exercise indicates that the exercise is also available as a video with step-by-step instructions. These videos are available on the website by following the instructions on the inside front cover. The study of probability stems from the analysis of certain games of chance, and it has found applications in most branches of science and engineering.

In this chapter the basic concepts of probability theory are presented.

In the study of probability, any process of observation is referred to as an