`and Statistics
`
`
`
`Second Edition
`
`Morris H. DeGroot
`
`Carnegie-Mellon University
`
`APPLE 1030
`Apple v. Speir
`IPR2022-01512
`
`Av
`
`v
`
`ADDISON-WESLEY PUBLISHING COMPANY
`
`Reading, Massachusetts/Menlo Park, California
`Don Mills, Ontario/Wokingham, England/Amsterdam/Sydney
`Singapore/Tokyo/Mexico City/Bogota/Santiago/San Juan
`
`APPLE 1030
`Apple v. Speir
`IPR2022-01512
`
`1
`
`
`
`This book is in the Addison-Wesley Series in Statistics.
`Frederick Mosteller Consulting Editor
`
`Library of Congress Cataloging in Publication Data
`
`DeGroot, Morris H.
`Probability and statistics.
`
`Bibliography: p.
`Includes index.
`2. Mathematical statistics.
`1. Probabilities.
`|. Title.
`519.2
`84-6269
`QA273.D35 1984
`ISBN 0-201-11366-X
`
`Reprinted with corrections, August 1987
`Copyright © 1975, 1986 by Addison-Wesley Publishing Company, Inc. All rights
`reserved. No part of this publication may be reproduced, stored in a retrieval
`system, oF transmitted, in any form or by any means, electronic, mechanical,
`photocopying, recording, or otherwise, without the prior written permission of the
`publisher. Printed in the United States of America. Published simultaneously in
`
`
`
`Canada.
`
`FGHIJ-MA-898
`
`2
`
`
`
`4.8. The Sample Mean
`
`227
`
`Var( X
`
`
`the Chebyshev Inequality. Let X be a random variable for which Var(X)
`
`bate
`exists. Then for any given number t > 0,
`(2)
`om the
`pe(|X— EU) | > 1) < YHOO,
`
`Proof Let Y =[X - E(X)/??. Then Pr(Y > 0)=1 and E(Y)= Var(X). By
`
`ying the Markov inequality to Y, we obtain the following result:
`pr(|X — E(X)| > th=Pr(¥>r?)< vat
`
`It can be seen from this proof that the Chebyshev inequality is simply a
`jal case of the Markov inequality. Therefore, the comments that were given
`lowing, the proof of the Markov inequality can be applied as well
`to the
`
`Chebyshev inequality. Because of their generality,
`these inequalities are very
`yseful. For example, if Var(X) = o* and we let
`¢ = 30,
`then the Chebyshev
`
`jnequality yields the result that
`
`
`
`In words, this result states that the probability that any given random variable
`ll differ from its mean by more than 3 standard deviations cannot exceed 1/9.
`
`This probability will actually be much smaller than 1/9 for many of the random
`mM. for
`ltype
`variables and distributions that will be discussed in this book. The Chebyshev
`
`7
`inequality is useful because of the fact that this probability must be 1/9 or less
`
`for every distribution. It can also be shown (see Exercise 3 at the end ofthis
`tion) that the upper boundin (2) is sharp in the sense that it cannot be made
`any smaller and still hold for a// distributions.
`
`
`Suppose that the random variables X,,..., X, form a random sampleofsize n
`
`from somedistribution for which the meanis p» and the varianceis o2. In other
`words, suppose that the random variables X,,..., X,, ate iid. and that each has
`
`Mean pw and variance o”. We shall let X,, represent the arithmetic average of the n
`
`observations in the sample. Thus,
`
`X,= *(X, tees +X).
`This random variable X,, is called the sample mean.
`
`Properties of the Sample Mean
`
`Pr(|X — E(X)| > 30) <
`
`olrR
`
`
`
`3
`
`
`
`228
`
`Expectation
`
`The mean and the variance of X, can easily be computed.It follows direcyy
`from the definition of X,, that
`:
`
`=,
`12
`1
`E(X,) = — De E(X) = 7 mea Be
`i=l
`
`Furthermore,since the variables X,,..., X,, are independent,
`
`Var(X,,) = 1var{ ¥ x,
`= 1S vax) = 4 -nor= =
`
`
`=F ar(X;)== 7
`
`i=1
`
`
`
`
`In words, the mean of X,, is equal to the mean of the distribution from whichthe
`random sample was drawn, butthe variance of X,, is only 1/n times the varia
`of that distribution. It follows that the probabilitydistribution of X,, will be m
`
`concentrated around the meanvalue p than wasthe original distribution, Ino
`words, the sample mean X, is morelikely to be close to p» thanis the value ofjust
`
`a single observation X, from the given distribution.
`These statements can be made more precise by applying the Chebyshey
`
`inequality to X,. Since E(X,}=p and Var(X,) = 07/n,
`it follows from the
`relation (2) that‘for any given number ¢ > 0,
`
`Example 1: Determining the Required Number of Observations. Suppose that a
`random sampleis to be taken from a distribution for which the value of the mear
`p is not known,but for whichit is known that the standard deviation o is 2 units.
`
`We shall determine how large the sample size must be in order to make the
`probability at least 0.99 that | X,, — | will be-less than 1 unit.
`
`Since o? = 4, it follows from the relation (3) that for any given sample size nm,
`
`Pr(|X,— ul > 1) < =.
`
`Since n must be chosen so that Pr(| X, — p| <1) > 0.99, it follows that n must
`be chosen so that 4/n < 0,01. Hence,it is required that n > 400.
`DO
`It should be emphasized thatthe use of the Chebyshev inequality in Example!
`guarantees that a sample for which n = 400 will be large enough to meet the
`specified probability requirements, regardless of the particular type of distribution’
`
`(|
`Pr X,-
`
`o2
`ul >t)
`on
`th< —.
`
`
`
`4
`
`
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