Mathematica Laboratories for Mathematical Statistics: Emphasizing Simulation and Computer Intensive Methods (ASA-SIAM Series on Statistics and Applied Probability)

281 Pages·2004·14.14 MB·English

by  Jenny A. Baglivo

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The correct bibliographic citation for this book is as follows: Baglivo, Jenny A.,
Mathematica Laboratories for Mathematical Statistics: Emphasizing Simulation and
Computer Intensive Methods, ASA-SIAM Series on Statistics and Applied Probability,
SIAM, Philadelphia, ASA, Alexandria, VA, 2005.
Copyright © 2005 by the American Statistical Association and the Society for Industrial
and Applied Mathematics.
10987654321
All rights reserved. Printed in the United States of America. No part of this book may
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Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688.
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This work was supported by Boston College through its faculty research programs and
by the National Science Foundation through its Division of Undergraduate Education
(NSFa DUE 9555178).
Library of Congress Cataloging-in-Publication Data
Baglivo, Jenny A. (Jenny Antoinette)
Mathematica laboratories for mathematical statistics : emphasizing simulation and
computer intensive methods / Jenny A. Baglivo.
p. cm. — (ASA-SIAM series on statistics and applied probability)
Includes bibliographical references and index.
ISBN 0-89871-566-0 (pbk.)
1. Mathematical statistics—Computer simulation. 2. Mathematica (Computer file)
I. Title. II. Series.
QA276.4.B34 2005
519.5'01'13-dc22 2004056579
slam.
is a registered trademark.
Disclaimer: This eBook does not include ancillary media that was packaged with the
printed version of the book.
Dedicatteotd h em emoryo f
Anthonayn dP hilomeBnaag livo
MarioFne rri
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Contents
Preface XV
1 IntroductProobrya biliCtoyn cepts 1
1.1 Definiti.o .n .s . . . 1
1.2 Kolmogoraoxvi om.s . . . . . 2
1.3 Countimnegt ho.d s. . .. .. 5
1.3.1 Permutatainodnc so mbinations 6
1.3.2 Partitisoentis.n .g 7
1.3.3 Generatfuinncgt io.n.s 9
1.4 Conditipornoabla bi.l i.t ..y . . 9
1.4.1 Lawo ftotpalr obability 11
1.4.2 Bayersu le. .. . . . . 11
1.5 Independeevnetn t.s . . .. . . . 12
1.5.1 Repeattreida lasn dm utuailn dependence 13
1.6 Laboratporroyb lem.s . . . .. .. . . . .. 13
1.6.1 LaboratoIrnytr:o ductcoornyc e.p ts 13
1.6.2 Additionparlo blneomt ebooks 13
2 DiscrePrtoeb abilDiitys tributions 15
2.1 Definiti.o .n .s . .. . .. . . . .. . .. . . . . 15
2.1.1 PDFa ndC DF fodri scrdeitset ributions 16
2.2 Univaridaitsetr ibut.i .o .n .s . . . .. . . . .. 17
2.2.1 ExamplDei:s cruentief odrims tribution 17
2.2.2 ExamplHey:p ergeomedtriisctr ibution 17
2.2.3 Distributrieolnastt eoBd e rnoulelxip erimen.t s 18
2.2.4 Simplrea ndosma mple.s . . 20
2.2.5 ExamplPeo:i ssdoins tribu.t .i .o .n . . . . 21
2.3 Joindti stributi.o .n .s . .. . . . . .. . .. . . . . . 22
2.3.1 Bivariadties tribumtiaorngsi;nd ails tributions 22
2.3.2 Conditidoinsatrli butiinodnesp;e nde.n .c. e 23
2.3.3 ExamplBei:v ariahtyep ergeomedtriisctr ibu.ti on 24
2.3.4 ExamplTer:i nomdiiasltr ibuti.o n 25
2.3.5 Surveayn alys.i .s . . . .. .. . 25
2.3.6 Discrmeutlet ivadriisatritbeu tions 26
2.3.7 Probabigleinteyr atfiunngc tions 26
vii
viii Contents
2.4 Laboratporroyb le.m s. . . . .. . . . 27
2.4.1 LaboratoDriys:c rmeotdee l.s 27
2.4.2 Additionparlo blneomt ebooks 27
3 ContinuoPruosb abilDiitsyt ributions 29
3.1 Definiti.o .n .s . .. . . . .. . . . . . .. . . . . . . . 29
3.1.1 PDFa ndC DF forc ontinuroaunsd ovma riables 29
3.1.2 Quantilpeesr;c enti.l.e.s. . 31
3.2 Univaridaitsetri buti.o.n.s. ....... . 31
3.2.1 ExamplUen:i fordmi stributio.n. 31
3.2.2 ExamplEex:p onentidails trib.u tion 32
3.2.3 Eulegra mmafu nctio.n . . . . . . 33
3.2.4 ExamplGeam:m ad istribu.ti o.n . 33
3.2.5 Distributrieolnastt eoPd o issporonc esses 34
3.2.6 ExamplCea:u chdyi stribu.t i.o .n . . . 35
3.2.7 ExamplNeo:rm alo rG aussidains tribution 35
3.2.8 ExamplLea:p ladcies tribut.i o.n . . .. . 36
3.2.9 Transforcmoinntg inuroaunsd ovmari ables 36
3.3 Joindti stribu.t .i .o. n .s . .. .. . . . . .. . .. 38
3.3.1 Bivaridaitset ribumtairognisnd;ai ls tributions 38
3.3.2 Conditidoinsatlr ibuitnidoenpse;n d.e nce 40
3.3.3 ExamplBei:v ariuantief odrims trib.u .t ion 41
3.3.4 ExamplBei:v arinaotrem adli stribu.t i.o n 41
3.3.5 Transformcionngt inuroaunsd ovmari ables 42
3.3.6 Continumouulst ivadriisatrtieb utions 43
3.4 Laboratporroyb lem.s. ........ . 44
3.4.1 LaboratoCroyn:t inumooudse ls 44
3.4.2 Additiopnraolb lneomt ebooks 44
4 MathematiEcxaple ctation 45
4.1 Definitiaonndps r oper.t ies 45
4.1.1 Discrdeitsetr ibut.i .o ns 45
4.1.2 Continudoiusst ributions 46
4.1.3 Proper.t i.e .s. . . . . 47
4.2 Meanv,a riansctea,n dadredv iation 48
4.2.1 Propert.i e.s . . . .. 48
4.2.2 Chebyshienve qual.i t.y 49
4.2.3 Markoivn equal.i .t .y 50
4.3 Functioonfst woo rm orer andovma riables 50
4.3.1 Propert.i.e.s. ... . 51
4.3.2 Covariacnocrer,e la.t.i.o.n. 51
.
4.3.3 Samplseu mmarie.s . . .. . .. 54
4.3.4 Conditieoxnpaelc tatrieognr;e ssion 55
4.4 Linefuanr ctioonfsr andovma riab.l .e. s . . 56
4.4.1 Independneonrtm arla ndovma riables 57
Contents ix
4.5 Laboratporroyb le.m s. . . . . .. . . . . . .. 58
4.5.1 LaboratoMrayt:h ematiecxapelc tation. 58
4.5.2 Additiopnraolb lneomt eboo.k.s. .. 58
5 LimiThteo rems 59
5.1 Definitio.n. s . . . .. . . . . . . .. . . . .. . . . . . 59
5.2 Lawo flargneu mber.s. . . . . . . . . . . . . . . . . . 60
5.2.1 ExamplMeo:n teC arleov aluatoifio nnt eg.r als 60
5.3 Centralli mtiht eore.m. . . . . 61
5.3.1 Continuciotryr ect.i on 62
5.3.2 Specicaals e.s . . . . 62
5.4 Momentg eneratifunngc tio.n. s 63
5.4.1 Methoodf m omengte neratifnugn cti.o ns 65
5.4.2 Relationtsoth hiecp e ntralli mtiht eorem 65
5.5 Laboratporroyb lem.s. . . . . .. . . . 66
5.5.1 LaboratoSruym:sa nda verages 66
5.5.2 Additiopnraolb lneomt eboo.k s 66
6 TransitiotnoS tatistics 69
6.1 Distributiroenlsa tteotd h en ormadli stribu.t ion 69
6.1.1 Chi-squdairsetr ibution 69
.
6.1.2 Studenttd istribut.i .o. n. . . . 70
6.1.3 F ratdiios tribut.i. o. n. . . . . 71
6.2 Randoms amplfreosm normald istributions 71
6.2.1 Samplmee ans,a mplvea ria.n c.e 72
6.2.2 Approximsattaen dardiozfat tihoesn am plem ean 73
6.2.3 Ratioo fs amplvea riances 74
6.3 Multinomeixaple rime.n .t. s. . . . . . 75
6.3.1 Multinomdiiasltr ibut.i. o. n. 75
6.3.2 Goodness-oKfn-ofiwtnm: o del 75
6.3.3 Goodness-oEfs-tfiimta:tm oedde l 77
6.4 Laboratporroyb lem.s . . . . . . . . . . . 79
6.4.1 LaboratoTrrya:n sittioso tna tis.t ics 79
6.4.2 Additionparolb lenmo tebooks 79
7 EstimatioThne ory 81
7.1 Definitio.n. s. . . . . .. . . . 81
.
7.2 Propertiesof p oinets timat.o. r. s 82
7.2.1 Biasu;n biaseesdti mator 82
7. 2.2 Efficiencfyo urn biaseesdt imators 82
7. 2.3 Means quareerrdo r. 83
7. .24 Consiste.n. c. y. . . . . . . 83
7.3 Interevsatli matio.n. . . . . .. . . . 84
7. 3.1 ExamplNeo:r madli stribution 84
7. 3.2 Approximianttee rvfaolmrse ans . 86
7.4 Methoodf m omentess timati.o. n . . 86
7.4.1 Singplaer ameetsetri mat.i. o. n 86