Table Of ContentJohn E. 1:-reund's
MArJ I
EMArJ~ICAL
S1A11 1 ICS
\"111 11 PP I I .,,A I I >
<.
f. I 111 11 I I> I I I <> :-..:
John E. Freund's Mathematical Statistics
with Applications
Irwin Miller Marylees Miller
Eighth Edition
ISBN 10: 1-292-02500-X
ISBN 13: 978-1-292-02500-1
Pearson Education Limited
Edinburgh Gate
Harlow
Essex CM20 2JE
England and Associated Companies throughout the world
Visit us on the World Wide Web at: www.pearsoned.co.uk
© Pearson Education Limited 2014
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the
prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom
issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS.
All trademarks used herein are the property of their respective owners. The use of any trademark
in this text does not vest in the author or publisher any trademark ownership rights in such
trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this
book by such owners.
ISBN 10: 1-292-02500-X
ISBN 13: 978-1-292-02500-1
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Printed in the United States of America
11122223332614703681351357731377911
P E A R S O N C U S T O M L I B R AR Y
Table of Contents
1. Introduction
Irwin Miller/Marylees Miller 1
2. Probability
Irwin Miller/Marylees Miller 21
3. Probability Distributions and Probability Densities
Irwin Miller/Marylees Miller 61
4. Mathematical Expectation
Irwin Miller/Marylees Miller 113
5. Special Probability Distributions
Irwin Miller/Marylees Miller 145
6. Special Probability Densities
Irwin Miller/Marylees Miller 177
7. Functions of Random Variables
Irwin Miller/Marylees Miller 207
8. Sampling Distributions
Irwin Miller/Marylees Miller 233
9. Decision Theory
Irwin Miller/Marylees Miller 261
10. Point Estimation
Irwin Miller/Marylees Miller 283
11. Interval Estimation
Irwin Miller/Marylees Miller 317
12. Hypothesis Testing
Irwin Miller/Marylees Miller 337
13. Tests of Hypothesis Involving Means, Variances, and Proportions
Irwin Miller/Marylees Miller 359
I
344444933346137939
14. Regression and Correlation
Irwin Miller/Marylees Miller 391
Appendix: Sums and Products
Irwin Miller/Marylees Miller 433
Appendix: Special Probability Distributions
Irwin Miller/Marylees Miller 437
Appendix: Special Probability Densities
Irwin Miller/Marylees Miller 439
Statistical Tables
Irwin Miller/Marylees Miller 443
Index 469
II
Introduction
1 Introduction 3 BinomialCoefficients
2 CombinatorialMethods 4 TheTheoryinPractice
1 Introduction
In recent years, the growth of statistics has made itself felt in almost every phase
of human activity. Statistics no longer consists merely of the collection of data and
theirpresentationinchartsandtables;itisnowconsideredtoencompassthescience
of basing inferences on observed data and the entire problem of making decisions
in the face of uncertainty. This covers considerable ground since uncertainties are
metwhenweflipacoin,whenadieticianexperimentswithfoodadditives,whenan
actuarydetermineslifeinsurancepremiums,whenaqualitycontrolengineeraccepts
orrejectsmanufacturedproducts,whenateachercomparestheabilitiesofstudents,
when an economist forecasts trends, when a newspaper predicts an election, and
evenwhenaphysicistdescribesquantummechanics.
It would be presumptuous to say that statistics, in its present state of devel-
opment, can handle all situations involving uncertainties, but new techniques are
constantly being developed and modern statistics can, at least, provide the frame-
work for looking at these situations in a logical and systematic fashion. In other
words, statistics provides the models that are needed to study situations involving
uncertainties, in the same way as calculus provides the models that are needed to
describe,say,theconceptsofNewtonianphysics.
Thebeginningsofthemathematicsofstatisticsmaybefoundinmid-eighteenth-
centurystudiesinprobabilitymotivatedbyinterestingamesofchance.Thetheory
thusdevelopedfor“headsortails”or“redorblack”soonfoundapplicationsinsit-
uationswheretheoutcomeswere“boyorgirl,”“lifeordeath,”or“passorfail,”and
scholars began to apply probability theory to actuarial problems and some aspects
of the social sciences. Later, probability and statistics were introduced into physics
by L. Boltzmann, J.Gibbs, and J.Maxwell, and by this century they have found
applications inallphases of human endeavor that insomewayinvolve anelement
of uncertainty or risk. The names that are connected most prominently with the
growthofmathematicalstatisticsinthefirsthalfofthetwentiethcenturyarethose
ofR.A.Fisher,J.Neyman,E.S.Pearson,andA.Wald.Morerecently,theworkof
R.Schlaifer,L.J.Savage, and othershasgiven impetustostatisticaltheories based
essentiallyonmethodsthatdatebacktotheeighteenth-centuryEnglishclergyman
ThomasBayes.
Mathematical statistics is a recognized branch of mathematics, and it can be
studiedforitsownsakebystudentsofmathematics.Today,thetheoryofstatisticsis
appliedtoengineering,physicsandastronomy,qualityassuranceandreliability,drug
development, public health and medicine, the design of agricultural or industrial
experiments, experimental psychology, and so forth. Those wishing to participate
FromChapter1ofJohnE.Freund’sMathematicalStatisticswithApplications,
EighthEdition.IrwinMiller,MaryleesMiller.Copyright©2014byPearsonEducation,Inc.
Allrightsreserved.
1
Introduction
in such applications or to develop new applications will do well to understand the
mathematicaltheoryofstatistics.Foronlythroughsuchanunderstandingcanappli-
cationsproceedwithouttheseriousmistakesthatsometimesoccur.Theapplications
are illustrated by means of examples and a separate set of applied exercises, many
oftheminvolvingtheuseofcomputers.Tothisend,wehaveaddedattheendofthe
chapteradiscussionofhowthetheoryofthechaptercanbeappliedinpractice.
We begin with a brief review of combinatorial methods and binomial
coefficients.
2 Combinatorial Methods
Inmanyproblemsofstatisticswemustlistallthealternativesthatarepossibleina
given situation, or at least determine how many different possibilities there are. In
connectionwiththelatter,weoftenusethefollowingtheorem,sometimescalledthe
basic principle of counting, the counting rule for compound events, or the rule for
themultiplicationofchoices.
THEOREM1.Ifanoperationconsistsoftwosteps,ofwhichthefirstcanbe
doneinn waysandforeachofthesethesecondcanbedoneinn ways,
1 2
thenthewholeoperationcanbedoneinn ·n ways.
1 2
Here,“operation”standsforanykindofprocedure,process,ormethodofselection.
Tojustifythistheorem,letusdefinetheorderedpair(x,y)tobetheoutcome
i j
that arises when the first step results in possibility x and the second step results in
i
possibility y. Then, the set of all possible outcomes is composed of the following
j
n ·n pairs:
1 2
(x ,y ),(x ,y ),...,(x ,y )
1 1 1 2 1 n2
(x ,y ),(x ,y ),...,(x ,y )
2 1 2 2 2 n2
...
...
...
(x ,y ),(x ,y ),...,(x ,y )
n1 1 n1 2 n1 n2
EXAMPLE1
Supposethatsomeonewantstogobybus,train,orplaneonaweek’svacationtoone
ofthefiveEastNorthCentralStates.Findthenumberofdifferentwaysinwhichthis
canbedone.
Solution
Theparticularstatecanbechoseninn = 5waysandthemeansoftransportation
1
can be chosen in n = 3 ways. Therefore, the trip can be carried out in 5·3 = 15
2
possible ways. If an actual listing of all the possibilities is desirable, a tree diagram
likethatinFigure1providesasystematicapproach.Thisdiagramshowsthatthere
are n = 5 branches (possibilities) for the number of states, and for each of these
1
branchestherearen = 3branches(possibilities)forthedifferentmeansoftrans-
2
portation.Itisapparentthatthe15possiblewaysoftakingthevacationarerepre-
sentedbythe15distinctpathsalongthebranchesofthetree.
2
Introduction
bus
train
plane
bus
o
Ohi train
plane
Indiana
bus
Illinois train
plane
Michigan
bus
W
is train
c
o
nsi plane
n
bus
train
plane
Figure1.Treediagram.
EXAMPLE2
How many possible outcomes are there when we roll a pair of dice, one red and
onegreen?
Solution
Thereddiecanlandinanyoneofsixways,andforeachofthesesixwaysthegreen
diecanalsolandinsixways.Therefore,thepairofdicecanlandin6·6=36ways.
Theorem1maybeextendedtocoversituationswhereanoperationconsistsof
twoormoresteps.Inthiscase,wehavethefollowingtheorem.
3
Introduction
THEOREM2. If an operation consists of k steps, of which the first can be
doneinn ways,foreachofthesethesecondstepcanbedoneinn ways,
1 2
foreachofthefirsttwothethirdstepcanbedoneinn ways,andsoforth,
3
thenthewholeoperationcanbedoneinn ·n ·...·n ways.
1 2 k
EXAMPLE3
Aqualitycontrolinspectorwishestoselectapartforinspectionfromeachoffour
differentbinscontaining4,3,5,and4parts,respectively.Inhowmanydifferentways
canshechoosethefourparts?
Solution
Thetotalnumberofwaysis4·3·5·4=240.
EXAMPLE4
In how many different ways can one answer all the questions of a true–false test
consistingof20questions?
Solution
Altogetherthereare
2·2·2·2· ... ·2·2=220 =1,048,576
different ways in which one can answer all the questions; only one of these corre-
spondstothecasewhereallthequestionsarecorrectandonlyonecorrespondsto
thecasewherealltheanswersarewrong.
Frequently,weareinterestedinsituationswheretheoutcomesarethedifferent
waysinwhichagroupofobjectscanbeorderedorarranged.Forinstance,wemight
wanttoknowinhowmanydifferentwaysthe24membersofaclubcanelectapresi-
dent,avicepresident,atreasurer,andasecretary,orwemightwanttoknowinhow
many different ways six persons can be seated around a table. Different arrange-
mentslikethesearecalledpermutations.
DEFINITION 1.PERMUTATIONS.Apermutationisadistinctarrangementofndiffer-
entelementsofaset.
EXAMPLE5
Howmanypermutationsarethereofthelettersa,b,andc?
Solution
The possible arrangements are abc, acb, bac, bca, cab, and cba, so the number of
distinctpermutationsissix.UsingTheorem2,wecouldhavearrivedatthisanswer
without actually listing the different permutations. Since there are three choices to
4
Introduction
select a letter for the first position, then two for the second position, leaving only
oneletterforthethirdposition,thetotalnumberofpermutationsis3·2·1=6.
Generalizingtheargumentusedintheprecedingexample,wefindthatndistinct
objectscanbearrangedinn(n−1)(n−2)·...·3·2·1differentways.Tosimplifyour
notation, we represent this product by the symbol n!, which is read “n factorial.”
Thus,1!=1,2!=2·1=2,3!=3·2·1 =6,4!=4·3·2·1 =24,5!=5·4·3·2·1 =
120,andsoon.Also,bydefinitionwelet0!=1.
THEOREM3.Thenumberofpermutationsofndistinctobjectsisn!.
EXAMPLE6
In how many different ways can the five starting players of a basketball team be
introducedtothepublic?
Solution
Thereare5!=5·4·3·2·1=120waysinwhichtheycanbeintroduced.
EXAMPLE7
The number of permutations of the four letters a, b, c, and d is 24, but what is the
numberofpermutationsifwetakeonlytwoofthefourlettersor,asitisusuallyput,
ifwetakethefourletterstwoatatime?
Solution
Wehavetwopositionstofill,withfourchoicesforthefirstandthenthreechoicesfor
thesecond.Therefore,byTheorem1,thenumberofpermutationsis4·3=12.
Generalizingtheargumentthatweusedintheprecedingexample,wefindthatn
distinct objects taken r at a time, for r>0, can be arranged in n(n−1)·...·
(n−r+1) ways. We denote this product by P , and we let P = 1 by definition.
n r n 0
Therefore,wecanstatethefollowingtheorem.
THEOREM4.Thenumberofpermutationsofndistinctobjectstakenrata
timeis
n!
P =
n r (n−r)!
forr=0,1,2,...,n.
Proof The formula P = n(n−1)·...·(n−r+1) cannot be used for
n r
r=0,butwedohave
n!
P = =1
n 0 (n−0)!
5
