Handbook of Statistical Distributions with Applications

346 Pages·2006·4.45 MB·English

Handbook of Statistical
Distributions with
Applications
K. Krishnamoorthy
University of Louisiana at Lafayette
U.S.A.
Boca Raton London New York
© 2006 by Taylor & Francis Group, LLC
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Library of Congress Cataloging-in-Publication Data
Krishnamoorthy, K. (Kalimuthu)
Handbook of statistical distributions with applications / K. Krishnamoorthy.
p. cm. -- (Statistics, a series of textbooks & monographs ; 188)
Includes bibliographical references and index.
ISBN 1-58488-635-8
1. Distribution (Probability theory)--Handbooks, manuals, etc. I. Title. II. Series:
Statistics, textbooks and monographs ; v. 188.
QA273.6.K75 2006
519.5--dc22 2006040297
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STATISTICS: Textbooks and Monographs
Recent Titles
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R. Rao and Gábor J. Székely
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V. Lakshmikantham
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and O. L. Gebizlioglu
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Muzaffar Shaikh
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© 2006 by Taylor & Francis Group, LLC
STATISTICS: Textbooks and Monographs
D. B. Owen
Founding Editor, 1972–1991
Associate Editors
Statistical Computing/ Multivariate Analysis
Nonparametric Statistics Professor Anant M. Kshirsagar
Professor William R. Schucany University of Michigan
Southern Methodist University
Quality Control/Reliability
Probability Professor Edward G. Schilling
Professor Marcel F. Neuts Rochester Institute of
University of Arizona Technology
Editorial Board
Applied Probability Statistical Process Improvement
Dr. Paul R. Garvey Professor G. Geoffrey Vining
The MITRE Corporation Virginia Polytechnic Institute
Economic Statistics Stochastic Processes
Professor David E. A. Giles Professor V. Lakshmikantham
University of Victoria Florida Institute of Technology
Experimental Designs Survey Sampling
Mr. Thomas B. Barker Professor Lynne Stokes
Rochester Institute of Southern Methodist University
Technology
Time Series
Multivariate Analysis Sastry G. Pantula
Professor Subir Ghosh North Carolina State University
University of California,
Riverside
Statistical Distributions
Professor N. Balakrishnan
McMaster University
© 2006 by Taylor & Francis Group, LLC
Contents
INTRODUCTION TO STATCALC
0.1 Introduction.......................................................1
0.2 Contents of StatCalc...............................................4
1 PRELIMINARIES
1.1 Random Variables and Expectations...............................9
1.2 Moments and Other Functions....................................12
1.2.1 Measures of Central Tendency.............................12
1.2.2 Moments..................................................12
1.2.3 Measures of Variability....................................13
1.2.4 Measures of Relative Standing.............................14
1.2.5 Other Measures ........................................... 14
1.2.6 Some Other Functions.....................................15
1.3 Some Functions Relevant to Reliability ........................... 15
1.4 Model Fitting .................................................... 16
1.4.1 Q–Q Plot..................................................17
1.4.2 The Chi-Square Goodness-of-Fit Test......................17
1.5 Methods of Estimation ........................................... 18
1.5.1 Moment Estimation.......................................18
1.5.2 Maximum Likelihood Estimation .......................... 19
1.6 Inference.........................................................19
1.6.1 Hypothesis Testing........................................19
1.6.2 Interval Estimation........................................23
1.7 Random Number Generation ..................................... 24
1.8 Some Special Functions...........................................25
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2 DISCRETE UNIFORM DISTRIBUTION
2.1 Description.......................................................29
2.2 Moments ........................................................ 30
3 BINOMIAL DISTRIBUTION
3.1 Description.......................................................31
3.2 Moments.........................................................32
3.3 Computing Table Values..........................................34
3.4 Test for the Proportion...........................................36
3.4.1 An Exact Test.............................................36
3.4.2 Power of the Exact Test...................................36
3.5 Confidence Intervals for the Proportion...........................38
3.5.1 An Exact Confidence Interval..............................38
3.5.2 Computing Exact Limits and Sample Size Calculation.....39
3.6 A Test for the Difference between Two Proportions...............40
3.6.1 An Unconditional Test.....................................40
3.6.2 Power of the Unconditional Test...........................41
3.7 Fisher’s Exact Test...............................................42
3.7.1 Calculation of p-Values....................................43
3.7.2 Exact Powers..............................................44
3.8 Properties and Results............................................45
3.8.1 Properties.................................................45
3.8.2 Relation to Other Distributions............................45
3.8.3 Approximations...........................................46
3.9 Random Number Generation ..................................... 46
3.10 Computation of Probabilities.....................................48
4 HYPERGEOMETRIC DISTRIBUTION
4.1 Description.......................................................51
4.2 Moments.........................................................52
4.3 Computing Table Values..........................................54
4.4 Point Estimation.................................................56
4.5 Test for the Proportion...........................................57
4.5.1 An Exact Test.............................................57
4.5.2 Power of the Exact Test...................................58
4.6 Confidence Intervals and Sample Size Calculation.................59
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4.6.1 Confidence Intervals.......................................59
4.6.2 Sample Size for Precision..................................60
4.7 A Test for the Difference between Two Proportions...............62
4.7.1 The Test..................................................62
4.7.2 Power Calculation.........................................63
4.8 Properties and Results............................................64
4.8.1 Recurrence Relations......................................64
4.8.2 Relation to Other Distributions............................64
4.8.3 Approximations...........................................64
4.9 Random Number Generation ..................................... 65
4.10 Computation of Probabilities.....................................66
5 POISSON DISTRIBUTION
5.1 Description.......................................................71
5.2 Moments.........................................................72
5.3 Computing Table Values..........................................74
5.4 Point Estimation.................................................75
5.5 Test for the Mean ................................................ 75
5.5.1 An Exact Test.............................................75
5.5.2 Powers of the Exact Test..................................76
5.6 Confidence Intervals for the Mean ................................ 77
5.6.1 An Exact Confidence Interval..............................77
5.6.2 Sample Size Calculation for Precision......................78
5.7 Test for the Ratio of Two Means..................................78
5.7.1 A Conditional Test........................................78
5.7.2 Powers of the Conditional Test ............................ 80
5.8 Confidence Intervals for the Ratio of Two Means..................81
5.9 A Test for the Difference between Two Means.....................81
5.9.1 An Unconditional Test.....................................82
5.9.2 Powers of the Unconditional Test..........................83
5.10 Model Fitting with Examples.....................................84
5.11 Properties and Results............................................86
5.11.1 Properties................................................86
5.11.2 Relation to Other Distributions...........................86
5.11.3 Approximations..........................................87
5.12 Random Number Generation ..................................... 87
5.13 Computation of Probabilities.....................................88
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6 GEOMETRIC DISTRIBUTION
6.1 Description.......................................................93
6.2 Moments.........................................................94
6.3 Computing Table Values..........................................94
6.4 Properties and Results............................................95
6.5 Random Number Generation ..................................... 96
7 NEGATIVE BINOMIAL DISTRIBUTION
7.1 Description.......................................................97
7.2 Moments.........................................................98
7.3 Computing Table Values.........................................100
7.4 Point Estimation................................................101
7.5 A Test for the Proportion ....................................... 101
7.6 Confidence Intervals for the Proportion..........................103
7.7 Properties and Results...........................................103
7.7.1 Properties................................................103
7.7.2 Relation to Other Distributions...........................104
7.8 Random Number Generation....................................104
7.9 A Computational Method for Probabilities.......................106
8 LOGARITHMIC SERIES DISTRIBUTION
8.1 Description......................................................107
8.2 Moments........................................................109
8.3 Computing Table Values.........................................109
8.4 Inferences.......................................................112
8.4.1 Point Estimation.........................................112
8.4.2 Interval Estimation.......................................112
8.5 Properties and Results...........................................113
8.6 Random Number Generation....................................113
8.7 A Computational Algorithm for Probabilities....................114
9 UNIFORM DISTRIBUTION
9.1 Description......................................................115
9.2 Moments........................................................116
9.3 Inferences.......................................................116
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9.4 Properties and Results...........................................117
9.5 Random Number Generation....................................117
10 NORMAL DISTRIBUTION
10.1 Description.....................................................119
10.2 Moments.......................................................123
10.3 Computing Table Values ....................................... 123
10.4 One-Sample Inference..........................................127
10.4.1 Point Estimation.......................................127
10.4.2 Test for the Mean and Power Computation.............128
10.4.3 Interval Estimation for the Mean.......................130
10.4.4 Test and Interval Estimation for the Variance...........132
10.5 Two-Sample Inference..........................................134
10.5.1 Inference for the Ratio of Variances.....................135
10.5.2 Inference for the Difference between Two Means
when the Variances Are Equal..........................136
10.5.3 Inference for the Difference between Two Means ....... 140
10.6 Tolerance Intervals.............................................142
10.6.1 Two-Sided Tolerance Intervals..........................142
10.6.2 One-Sided Tolerance Limits.............................143
10.6.3 Equal-Tail Tolerance Intervals..........................145
10.6.4 Simultaneous Hypothesis Testing for Quantiles..........146
10.6.5 Tolerance Limits for One-Way Random Effects Model...147
10.7 Properties and Results.........................................149
10.8 Relation to Other Distributions.................................150
10.9 Random Number Generation...................................151
10.10 Computing the Distribution Function...........................152
11 CHI-SQUARE DISTRIBUTION
11.1 Description.....................................................155
11.2 Moments.......................................................156
11.3 Computing Table Values ....................................... 157
11.4 Applications....................................................157
11.5 Properties and Results.........................................158
11.5.1 Properties..............................................158
11.5.2 Relation to Other Distributions.........................159
11.5.3 Approximations........................................160
11.6 Random Number Generation...................................161
11.7 Computing the Distribution Function...........................161
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12 F DISTRIBUTION
12.1 Description.....................................................163
12.2 Moments.......................................................165
12.3 Computing Table Values ....................................... 165
12.4 Properties and Results.........................................166
12.4.1 Identities...............................................166
12.4.2 Relation to Other Distributions.........................166
12.4.3 Series Expansions.......................................167
12.4.4 Approximations........................................168
12.5 Random Number Generation...................................168
12.6 A Computational Method for Probabilities ..................... 169
13 STUDENT’S t DISTRIBUTION
13.1 Description.....................................................171
13.2 Moments.......................................................172
13.3 Computing Table Values ....................................... 173
13.4 Distribution of the Maximum of Several |t| Variables ........... 173
13.4.1 An Application.........................................174
13.4.2 Computing Table Values................................175
13.4.3 An Example............................................175
13.5 Properties and Results.........................................176
13.5.1 Properties..............................................176
13.5.2 Relation to Other Distributions.........................176
13.5.3 Series Expansions for Cumulative Probability...........177
13.5.4 An Approximation ..................................... 178
13.6 Random Number Generation...................................178
13.7 A Computational Method for Probabilities ..................... 178
14 EXPONENTIAL DISTRIBUTION
14.1 Description.....................................................179
14.2 Moments.......................................................180
14.3 Computing Table Values ....................................... 180
14.4 Inferences......................................................181
14.5 Properties and Results.........................................182
14.5.1 Properties..............................................182
14.5.2 Relation to Other Distributions.........................182
14.6 Random Number Generation...................................183
© 2006 by Taylor & Francis Group, LLC