Table Of ContentBinomial Distribution Handbook
for Scientists and Engineers
Elart von Collani
Klaus Drăger
Binomial Distribution Handbook
for Scientists and Engineers
Springer Science+Business Media, LLC
Elart von Collani
Klaus Drăger
University of Wiirzburg
Sanderring 2
D-97070 Wiirzburg
Germany
Library of Congress Cataloging-in-Publication Data
von Collani, Elart, 1944-
Binomial distribution handbook for scientists and engineers I Elart von Collani, Klaus Drăger.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4612-6666-2 ISBN 978-1-4612-0215-8 (eBook)
DOI 10.1007/978-1-4612-0215-8
1. Binomial distribution. 1. Drăger, Klaus.
QA273.6 .V65 2001
5 12 .2'4-dc2 1 00-046821
CIP
Printed on acid-free paper.
© 2001 Springer Science+Business Media New York
Originally published by Birkhăuser Boston in 2001
Softcover reprint of the hardcover 1s t edition 200 1
AlI rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in
connection with reviews or scho1arly analysis. Use in connection with any form of information storage
and retrieval, electronic adaptation, computer software, or by similar or dissimi1ar methodology now
known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe former
are not especially identified, is not to be taken as a sign that such names, as understood by the Trade
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Additional material to this book can be downloaded from http://extras.springer.com
ISBN 978-1-4612-6666-2 SPIN 10723723
Production managed by Louise Farkas; manufacturing supervised by Erica Bresler.
Typeset by the author in LaTeX.
9 8 765 4 3 2 1
To Claudia and Machiko
Contents
Preface xiii
I Introduction 1
1 Stochastics 3
1.1 The Science of Stochastics 3
1.2 Historical Remarks .... 8
1.3 Measurement Procedure and
Measurement Range ..... 10
2 Models Related to the Probability of an Event 12
2.1 The Concept of Probability 12
2.2 Random Variables and Data 16
2.3 The Model. . . . . . . . . 17
2.4 The Random Sample . . . . 19
2.5 The Binomial Distribution . 21
2.6 The Hypergeometric Distribution 27
2.7 Measuring in the Measurement Range. 28
3 Traditional Estimation Procedures 32
3.1 Theory of Estimation . . . . . . . . 32
3.1.1 Neyman's Approach .... 32
3.1.2 Point and Interval Estimation 39
3.2 Interval Estimator for a Probability . 42
3.3 The Relative Frequency X . . . . . . 46
3.4 Measurement Procedures Based on the Relative
Frequency . . . . . . . . . . . . . . . . . . . 48
Vll
Vlll CONTENTS
3.4.1 Traditional Measurement Procedures 49
3.4.2 Approximate Interval Estimators .. 52
II Theory 57
4 Measurement and Prediction Procedures 59
4.1 The Problem Revisited . 59
4.2 Measurement &Prediction Space 62
4.2.1 Measurement & Prediction Space for (p, X) 63
4.2.2 Measurement &PredictionSpacefor (p, Xs) 67
4.3 ,B-Measurement & Prediction. . . 69
4.3.1 The ,B-Measurement & Prediction
Space for (p, X) . . . . 70
4.3.2 The ,B-Measurement &Prediction
Space for (p, Xs) .. 72
4 3 3 The Relation Between M~,n) and M(!3,n] 73
. . X,p Xs,p
4.4 Quality of a Measurement Procedure 75
4.4.1 Quality of the 13-Measurement & Predic-
tion Space for (X,p) .. . . 75
4.4.2 Quality of the ,B-Measurement & Predic
tion Space for (Xs,p) ... 78
4.5 Neyman ,B-Measurement Procedure ... 79
4.6 Determination of Neyman Measurement Proce
dures . 81
4.7 Limiting Quality of Neyman Procedures 83
4.8 ,B-Measurement & Prediction Space for a Large
Sample Size 85
4.9 Illustrative Example 87
X .
4.9.1 Prediction Procedure Based on 88
X
4.9.2 Measurement Procedure Based on 91
4.9.3 ,B-Measurement & Prediction Space
Based on (p, X). ... 91
4.9.4 Prediction & Measurement Procedure Based
on Xs . . .. .. 93
4.9.5 Traditional Measurement Procedure. 94
CONTENTS IX
5 Complete Measurement Procedures 97
5.1 Point Estimators . . . . . . . . 97
5.2 Traditional Point Estimator .. 99
5.3 Measurement Space Conformity · 100
5.4 The ,B-Estimator . · 105
5.4.1 The Minimum MSE-,B-Estimator .106
5.5 The Midpoint ,B-Estimator . · 111
5.6 Illustrative Example for ,B-Estimators .. .113
5.6.1 Conditional Mean Squared Error .119
6 Exclusion Procedures 120
6.1 Rejecting a Null Hypothesis . · 120
6.1.1 The Exclusion Procedure as a Decision
Procedure . · 123
6.2 a-Exclusion Space . · 124
6.3 Quality of an a-Exclusion Procedure · 127
6.4 Neyman Exclusion Procedure · 129
6.5 Determination of Neyman
Exclusion Procedures . . . . . . . . . . . . . . . · 129
6.6 Illustrative Example for an
a-Exclusion Procedure . · 131
6.6.1 The Measurement Space and
the Exclusion Procedure · 136
7 Comparison Procedures 145
7.1 The Alternatives . · 145
7.2 The Decision Function . · 147
7.3 The aI, ... ,am-Comparison Space. · 148
7.4 Quality of a Comparison Procedure · 151
7.5 Neyman Comparison Procedure .. · 151
7.6 The Special Case of an aI, az-Comparison Pro-
cedure 153
7.7 Determination of Neyman
Comparison Procedures 155
7.8 Illustrative Examples for an aI,az-Comparison
Procedure . . . . . . . . . . . . . . . . . . . . . . 157
x CONTENTS
III Introduction to the Tables 163
8 Measurement Intervals 165
8.1 Tables for Measurement Intervals 165
8.2 The Printed Tables 166
8.3 Illustrative Examples for the Use of the Printed
Tables . . . . . . . . . . . . . . . . . . . 167
8.4 The Tables on CD-ROM . . . . . . . . . 169
8.4.1 Input Mask I for Measurement
Intervals . · 173
8.5 Illustrative Example for the Use of the
CD-ROM Tables . · 175
8.6 Effects of a Restricted Measurement Space · 177
9 Prediction Regions 182
9.1 Tables of Prediction Regions on CD-ROM . 182
9.1.1 Input Mask II for Prediction Regions . 183
9.2 Illustrative Examples for Determining a
Prediction Region . . . . . . . . . . . 185
9.3 Howith Positive Lebesgue Measure . 187
IV Application 191
10 Measuring a Probability 193
10.1 The Actual Value of a Probability. .194
10.1.1 Algorithm . · 194
10.1.2 Examples . · 194
10.2 The Difference of Two Probabilities .200
10.3 Ratio of Poisson Parameters . .203
10.4 Point Estimator . .206
10.5 Determination of Point Estimates .210
11 Excluding a Probability 212
11.1 Special Exclusion Procedures .213
11.1.1 Binomial Test . . .213
11.1.2 Sign Test .... .218
11.1.3 Test of McNemar .222
CONTENTS Xl
11.1.4 Chapman Test .225
12 Comparing Probabilities 227
12.1 Comparisons by Means of the Tables .227
12.2 Illustrative Examples for
aI,a2-Comparison Procedures .... .231
V Tables 237
Glossary 347
References 351
Index 355
Preface
In 1986, the eminent statistician John NeIder [23] noted in the
Journal of the Royal Statistical Society (Series A ): "If statis
tics is an applied field and not a minor branch of mathematics,
then more than ninety-nine percent of the published papers are
useless exercises." Evidently, statistics is an applied field, as its
application has pervaded all branches ofscience and technology,
and therefore the question of how to get rid of the huge pile of
rubbish that hides the useful things must be discussed.
One possibility is to return to the roots and make an attempt
to reformulate the problems aimed at disclosing aberrations,
show desirable directions, and develop easy-to-apply methods
relevant for application.
This handbook deals with the oldest problem in stochastics
and goes back to its deepest roots in order to elaborate on the
old problem from a different point ofview, thus addressing pro
fessional and academic statisticians. New solutions for the old
problem are obtained and subsequently presented for the user
without assuming mathematical skills and a deep knowledge of
statistics. The aim is to offer statistical methods to the poten
tial user in a comparable way as sophisticated technical devices
are made available, which can be used by any layman without
understanding the underlying physical or chemical principles.
There were two primary motives for writing this handbook.
The first one resulted from a consulting project in the course
of which proportions had to be estimated. We noticed that
the recommended methods are difficult to handle for practition
ers and, moreover, they are inaccurate and yield often useless
results. This made us think about the "quality" of statistical
Xlll
