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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A new edition of the trusted guide on commonly used statistical distributions Fully updated to reflect the latest developments on the topic, Statistical Distributions, Fourth Edition continues to serve as an authoritative guide on the application of statistical methods to research across various disciplines. The book provides a concise presentation of popular statistical distributions along with the necessary knowledge for their successful use in data modeling and analysis. Following a basic introduction, forty popular distributions are outlined in individual chapters that are complete with related facts and formulas. Reflecting the latest changes and trends in statistical distribution theory, the Fourth Edition features: * A new chapter on queuing formulas that discusses standard formulas that often arise from simple queuing systems * Methods for extending independent modeling schemes to the dependent case, covering techniques for generating complex distributions from simple distributions * New coverage of conditional probability, including conditional expectations and joint and marginal distributions * Commonly used tables associated with the normal (Gaussian), student-t, F and chi-square distributions * Additional reviewing methods for the estimation of unknown parameters, such as the method of percentiles, the method of moments, maximum likelihood inference, and Bayesian inference Statistical Distributions, Fourth Edition is an excellent supplement for upper-undergraduate and graduate level courses on the topic. It is also a valuable reference for researchers and practitioners in the fields of engineering, economics, operations research, and the social sciences who conduct statistical analyses.
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Seitenzahl: 177
Veröffentlichungsjahr: 2011
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Contents
Cover
Title Page
Copyright
Dedication
Preface
Chapter 1: Introduction
Chapter 2: Terms and Symbols
2.1 Probability, Random Variable, Variate, and Number
2.2 Range, Quantile, Probability Statement, and Domain
2.3 Distribution Function and Survival Function
2.4 Inverse Distribution Function and Inverse Survival Function
2.5 Probability Density Function and Probability Function
2.6 Other Associated Functions and Quantities
Chapter 3: General Variate Relationships
3.1 Introduction
3.2 Function of a Variate
3.3 One-to-One Transformations and Inverses
3.4 Variate Relationships Under One-to-One Transformation
3.5 Parameters, Variate, and Function Notation
3.6 Transformation of Location and Scale
3.7 Transformation from the Rectangular Variate
3.8 Many-to-One Transformations
Chapter 4: Multivariate Distributions
4.1 Joint Distributions
4.2 Marginal Distributions
4.3 Independence
4.4 Conditional Distributions
4.5 Bayes' Theorem
4.6 Functions of a Multivariate
Chapter 5: Stochastic Modeling
5.1 Introduction
5.2 Independent Variates
5.3 Mixture Distributions
5.4 Skew-Symmetric Distributions
5.5 Distributions Characterized by Conditional Skewness
5.6 Dependent Variates
Chapter 6: Parameter Inference
6.1 Introduction
6.2 Method of Percentiles Estimation
6.3 Method of Moments Estimation
6.4 Maximum Likelihood Inference
6.5 Bayesian Inference
Chapter 7: Bernoulli Distribution
7.1 Random Number Generation
7.2 Curtailed Bernoulli Trial Sequences
7.3 URN Sampling Scheme
7.4 Note
Chapter 8: Beta Distribution
8.1 Notes on Beta and Gamma Functions
8.2 Variate Relationships
8.3 Parameter Estimation
8.4 Random Number Generation
8.5 Inverted Beta Distribution
8.6 Noncentral Beta Distribution
8.7 Beta Binomial Distribution
Chapter 9: Binomial Distribution
9.1 Variate Relationships
9.2 Parameter Estimation
9.3 Random Number Generation
Chapter 10: Cauchy Distribution
10.1 Note
10.2 Variate Relationships
10.3 Random Number Generation
10.4 Generalized Form
Chapter 11: Chi-Squared Distribution
11.1 Variate Relationships
11.2 Random Number Generation
11.3 Chi Distribution
Chapter 12: Chi-Squared (Noncentral) Distribution
12.1 Variate Relationships
Chapter 13: Dirichlet Distribution
13.1 Variate Relationships
13.2 Dirichlet Multinomial Distribution
Chapter 14: Empirical Distribution Function
14.1 Estimation from Uncensored Data
14.2 Estimation from Censored Data
14.3 Parameter Estimation
14.4 Example
14.5 Graphical Method for the Modified Order-Numbers
14.6 Model Accuracy
Chapter 15: Erlang Distribution
15.1 Variate Relationships
15.2 Parameter Estimation
15.3 Random Number Generation
Chapter 16: Error Distribution
16.1 Note
16.2 Variate Relationships
Chapter 17: Exponential Distribution
17.1 Note
17.2 Variate Relationships
17.3 Parameter Estimation
17.4 Random Number Generation
Chapter 18: Exponential Family
18.1 Members Of The Exponential Family
18.2 Univariate One-Parameter Exponential Family
18.3 Parameter Estimation
18.4 Generalized Exponential Distributions
Chapter 19: Extreme Value (Gumbel) Distribution
19.1 Note
19.2 Variate Relationships
19.3 Parameter Estimation
19.4 Random Number Generation
Chapter 20: F (Variance Ratio) or Fisher–Snedecor Distribution
20.1 Variate Relationships
Chapter 21: F (Noncentral) Distribution
21.1 Variate Relationships
Chapter 22: Gamma Distribution
22.1 Variate Relationships
22.2 Parameter Estimation
22.3 Random Number Generation
22.4 Inverted Gamma Distribution
22.5 Normal Gamma Distribution
22.6 Generalized Gamma Distribution
Chapter 23: Geometric Distribution
23.1 Notes
23.2 Variate Relationships
23.3 Random Number Generation
Chapter 24: Hypergeometric Distribution
24.1 Note
24.2 Variate Relationships
24.3 Parameter Estimation
24.4 Random Number Generation
24.5 Negative Hypergeometric Distribution
24.6 Generalized Hypergeometric Distribution
Chapter 25: Inverse Gaussian (Wald) Distribution
25.1 Variate Relationships
25.2 Parameter Estimation
Chapter 26: Laplace Distribution
26.1 Variate Relationships
26.2 Parameter Estimation
26.3 Random Number Generation
Chapter 27: Logarithmic Series Distribution
27.1 Variate Relationships
27.2 Parameter Estimation
Chapter 28: Logistic Distribution
28.1 Notes
28.2 Variate Relationships
28.3 Parameter Estimation
28.4 Random Number Generation
Chapter 29: Lognormal Distribution
29.1 Variate Relationships
29.2 Parameter Estimation
29.3 Random Number Generation
Chapter 30: Multinomial Distribution
30.1 Variate Relationships
30.2 Parameter Estimation
Chapter 31: Multivariate Normal (Multinormal) Distribution
31.1 Variate Relationships
31.2 Parameter Estimation
Chapter 32: Negative Binomial Distribution
32.1 Note
32.2 Variate Relationships
32.3 Parameter Estimation
32.4 Random Number Generation
Chapter 33: Normal (Gaussian) Distribution
33.1 Variate Relationships
33.2 Parameter Estimation
33.3 Random Number Generation
33.4 Truncated Normal Distribution
33.5 Variate Relationships
Chapter 34: Pareto Distribution
34.1 Note
34.2 Variate Relationships
34.3 Parameter Estimation
34.4 Random Number Generation
Chapter 35: Poisson Distribution
35.1 Note
35.2 Variate Relationships
35.3 Parameter Estimation
35.4 Random Number Generation
Chapter 36: Power Function Distribution
36.1 Variate Relationships
36.2 Parameter Estimation
36.3 Random Number Generation
Chapter 37: Power Series (Discrete) Distribution
37.1 Note
37.2 Variate Relationships
37.3 Parameter Estimation
Chapter 38: Queuing Formulas
38.1 Characteristics of Queuing Systems and Kendall-Lee Notation
38.2 Definitions, Notation, and Terminology
38.3 General Formulas
38.4 Some Standard Queuing Systems
Chapter 39: Rayleigh Distribution
39.1 Variate Relationships
39.2 Parameter Estimation
Chapter 40: Rectangular (Uniform) Continuous Distribution
40.1 Variate Relationships
40.2 Parameter Estimation
40.3 Random Number Generation
Chapter 41: Rectangular (Uniform) Discrete Distribution
41.1 General Form
41.2 Parameter Estimation
Chapter 42: Student's t Distribution
42.1 Variate Relationships
42.2 Random Number Generation
Chapter 43: Student's t (Noncentral) Distribution
43.1 Variate Relationships
Chapter 44: Triangular Distribution
44.1 Variate Relationships
44.2 Random Number Generation
Chapter 45: von Mises Distribution
45.1 Note
45.2 Variate Relationships
45.3 Parameter Estimation
Chapter 46: Weibull Distribution
46.1 Note
46.2 Variate Relationships
46.3 Parameter Estimation
46.4 Random Number Generation
46.5 Three-Parameter Weibull Distribution
46.6 Three-Parameter Weibull Random Number Generation
46.7 Bi-Weibull Distribution
46.8 Five-Parameter Bi-Weibull Distribution
46.9 Weibull Family
Chapter 47: Wishart (Central) Distribution
47.1 Note
47.2 Variate Relationships
Chapter 48: Statistical Tables
Bibliography
Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Statistical distributions. – 4th ed. / Catherine Forbes... [et al.].
p. cm. – (Wiley series in probability and statistics)
Includes bibliographical references and index.
ISBN 978-0-470-39063-4 (pbk.)
1. Distribution (Probability theory) I. Forbes, Catherine.
QA273.6.E92 2010
519.2'4–dc22
2009052131
TO Jeremy and Elana Forbes
Caitlin and Eamon Evans
Tina Hastings
Eileen Peacock
Preface
This revised handbook provides a concise summary of the salient facts and formulas relating to 40 major probability distributions, together with associated diagrams that allow the shape and other general properties of each distribution to be readily appreciated.
In the introductory chapters the fundamental concepts of the subject are covered with clarity, and the rules governing the relationships between variates are described. Extensive use is made of the inverse distribution function and a definition establishes a variate as a generalized form of a random variable. A consistent and unambiguous system of nomenclature can thus be developed, with chapter summaries relating to individual distributions.
Students, teachers, and practitioners for whom statistics is either a primary or secondary discipline will find this book of great value, both for factual references and as a guide to the basic principles of the subject. It fulfills the need for rapid access to information that must otherwise be gleaned from many scattered sources.
The first version of this book, written by N. A. J. Hastings and J. B. Peacock, was published by Butterworths, London, 1975. The second edition, with a new author, M. A. Evans, was published by John Wiley & Sons in 1993, with a third edition by the same authors published by John Wiley & Sons in 2000. This fourth edition sees the addition of a new author, C. S. Forbes. Catherine Forbes holds a Ph.D. in Mathematical Statistics from The Ohio State University, USA, and is currently Senior Lecturer at Monash University, Victoria, Australia. Professor Merran Evans is currently Pro Vice-Chancellor, Planning and Quality at Monash University and obtained her Ph.D. in Econometrics from Monash University. Dr. Nicholas Hastings holds a Ph.D. in Operations Research from the University of Birmingham. Formerly Mount Isa Mines Professor of Maintenance Engineering at Queensland University of Technology, Brisbane, Australia, he is currently Director and Consultant in physical asset management, Albany Interactive Pty Ltd. Dr. Brian Peacock has a background in ergonomics and industrial engineering which have provided a foundation for a long career in industry and academia, including 18 years in academia, 15 years with General Motors' vehicle design and manufacturing organizations, and 4 years as discipline coordinating scientist for the National Space Biomedical Institute/NASA. He is a licensed professional engineer, a licensed private pilot, a certified professional ergonomist, and a fellow of both the Ergonomics and Human Factors Society (UK) and the Human Factors and Ergonomics Society (USA). He recently retired as a professor in the Department of Safety Science at Embry Riddle Aeronautical University, where he taught classes in system safety and applied ergonomics.
The authors gratefully acknowledge the helpful suggestions and comments made by Harry Bartlett, Jim Conlan, Benoit Dulong, Alan Farley, Robert Kushler, Jerry W. Lewis, Allan T. Mense, Grant Reinman, and Dimitris Ververidis.
Catherine Forbes Merran Evans Nicholas Hastings Brian Peacock
Chapter 1
Introduction
The number of puppies in a litter, the life of a light bulb, and the time to arrival of the next bus at a stop are all examples of random variables encountered in everyday life. Random variables have come to play an important role in nearly every field of study: in physics, chemistry, and engineering, and especially in the biological, social, and management sciences. Random variables are measured and analyzed in terms of their statistical and probabilistic properties, an underlying feature of which is the distribution function. Although the number of potential distribution models is very large, in practice a relatively small number have come to prominence, either because they have desirable mathematical characteristics or because they relate particularly well to some slice of reality or both.
This book gives a concise statement of leading facts relating to 40 distributions and includes diagrams so that shapes and other general properties may readily be appreciated. A consistent system of nomenclature is used throughout. We have found ourselves in need of just such a summary on frequent occasions—as students, as teachers, and as practitioners. This book has been prepared and revised in an attempt to fill the need for rapid access to information that must otherwise be gleaned from scattered and individually costly sources.
In choosing the material, we have been guided by a utilitarian outlook. For example, some distributions that are special cases of more general families are given extended treatment where this is felt to be justified by applications. A general discussion of families or systems of distributions was considered beyond the scope of this book. In choosing the appropriate symbols and parameters for the description of each distribution, and especially where different but interrelated sets of symbols are in use in different fields, we have tried to strike a balance between the various usages, the need for a consistent system of nomenclature within the book, and typographic simplicity. We have given some methods of parameter estimation where we felt it was appropriate to do so. References listed in the Bibliography are not the primary sources but should be regarded as the first “port of call”.
In addition to listing the properties of individual variates we have considered relationships between variates. This area is often obscure to the nonspecialist. We have also made use of the inverse distribution function, a function that is widely tabulated and used but rarely explicitly defined. We have particularly sought to avoid the confusion that can result from using a single symbol to mean here a function, there a quantile, and elsewhere a variate.
Building on the three previous editions, this fourth edition documents recent extensions to many of these probability distributions, facilitating their use in more varied applications. Details regarding the connection between joint, marginal, and conditional probabilities have been included, as well as new chapters (Chapters 5 and 6) covering the concepts of statistical modeling and parameter inference. In addition, a new chapter (Chapter 38) detailing many of the existing standard queuing theory results is included. We hope the new material will encourage readers to explore new ways to work with statistical distributions.
Chapter 2
Terms and Symbols
2.1 Probability, Random Variable, Variate, and Number
Probabilistic Experiment
A probabilistic experiment is some occurrence such as the tossing of coins, rolling dice, or observation of rainfall on a particular day where a complex natural background leads to a chance outcome.
Sample Space
The set of possible outcomes of a probabilistic experiment is called the sample, event, or possibility space. For example, if two coins are tossed, the sample space is the set of possible results HH, HT, TH, and TT, where H indicates a head and T a tail.
Random Variable
A random variable is a function that maps events defined on a sample space into a set of values. Several different random variables may be defined in relation to a given experiment. Thus, in the case of tossing two coins the number of heads observed is one random variable, the number of tails is another, and the number of double heads is another. The random variable “number of heads” associates the number 0 with the event TT, the number 1 with the events TH and HT, and the number 2 with the event HH. Figure 2.1 illustrates this mapping.
Figure 2.1 The random variable “number of heads”.
Variate
In the discussion of statistical distributions it is convenient to work in terms of variates. A variate is a generalization of the idea of a random variable and has similar probabilistic properties but is defined without reference to a particular type of probabilistic experiment. A variate is the set of all random variables that obey a given probabilistic law. The number of heads and the number of tails observed in independent coin tossing experiments are elements of the same variate since the probabilistic factors governing the numerical part of their outcome are identical.
A multivariate is a vector or a set of elements, each of which is a variate. A matrix variate is a matrix or two-dimensional array of elements, each of which is a variate. In general, dependencies may exist between these elements.
Random Number
A random number associated with a given variate is a number generated at a realization of any random variable that is an element of that variate.
2.2 Range, Quantile, Probability Statement, and Domain
Range
Let X denote a variate and let be the set of all (real number) values that the variate can take. The set is the range of X. As an illustration (illustrations are in terms of random variables) consider the experiment of tossing two coins and noting the number of heads. The range of this random variable is the set heads, since the result may show zero, one, or two heads. (An alternative common usage of the term range refers to the largest minus the smallest of a set of variate values.)
Quantile
For a general variate X let x (a real number) denote a general element of the range . We refer to x as the quantile of X. In the coin tossing experiment referred to previously, heads; that is, x is a member of the set heads.
Probability Statement
Let mean “the value realized by the variate X is x.” Let mean “the probability that the value realized by the variate X is less than or equal to x.”
Probability Domain
Let α (a real number between 0 and 1) denote probability. Let be the set of all values (of probability) that can take. For a continuous variate, is the line segment [0, 1]; for a discrete variate it will be a subset of that segment. Thus is the probability domain of the variate X.
In examples we shall use the symbol X to denote a random variable. Let X be the number of heads observed when two coins are tossed. We then have
and hence .
2.3 Distribution Function and Survival Function
Distribution Function
The distribution function F (or more specifically ) associated with a variate X maps from the range into the probability domain or [0, 1] and is such that
(2.1)
The function F(x) is nondecreasing in x and attains the value unity at the maximum of x. Figure 2.2 illustrates the distribution function for the number of heads in the experiment of tossing two coins. Figure 2.3 illustrates a general continuous distribution function and Figure 2.4 a general discrete distribution function.
Figure 2.2 The distribution function F: or for the random variable, “number of heads”.
Figure 2.3 Distribution function and inverse distribution function for a continuous variate.
Figure 2.4 Distribution function and inverse distribution function for a discrete variate.
Survival Function
The survival functionS(x) is such that
2.4 Inverse Distribution Function and Inverse Survival Function
For a distribution function F, mapping a quantile x into a probability α, the quantile function or inverse distribution function G performs the corresponding inverse mapping from α into x. Thus for , the following statements hold:
(2.2)
(2.3)
(2.4)
where is the quantile such that the probability that the variate takes a value less than or equal to it is is the percentile.
Figures 2.2, 2.3, and 2.4 illustrate both distribution functions and inverse distribution functions, the difference lying only in the choice of independent variable.
For the two-coin tossing experiment the distribution function F and inverse distribution function G of the number of heads are as follows:
Inverse Survival Function
The inverse survival function Z is a function such that is the quantile, which is exceeded with probability α. This definition leads to the following equations:
Inverse survival functions are among the more widely tabulated functions in statistics. For example, the well-known chi-squared tables are tables of the quantile x as a function of the probability level α and a shape parameter, and hence are tables of the chi-squared inverse survival function.
2.5 Probability Density Function and Probability Function
A probability density function, f(x), is the first derivative coefficient of a distribution function, F(x), with respect to x (where this derivative exists).
For a given continuous variate X the area under the probability density curve between two points in the range of X is equal to the probability that an as-yet unrealized random number of X will lie between and . Figure 2.5 illustrates this. Figure 2.6 illustrates the relationship between the area under a probability density curve and the quantile mapped by the inverse distribution function at the corresponding probability value.
Figure 2.5 Probability density function.
Figure 2.6 Probability density function illustrating the quantile corresponding to a given probability is the inverse distribution function.
A discrete variate takes discrete values x with finite probabilities f(x). In this case f(x) is the probability function, also called the probability mass function.
2.6 Other Associated Functions and Quantities
In addition to the functions just described, there are many other functions and quantities that are associated with a given variate. A listing is given in Table 2.1 relating to a general variate that may be either continuous or discrete. The integrals in Table 2.1 are Stieltjes integrals, which for discrete variates become ordinary summations, so
Table 2.2 gives some general relationships between moments, and Table 2.3 gives our notation for values, mean, and variance for samples.
Table 2.1 Functions and Related Quantities for a General Variate (X Denotes a Variate, x a Quantile, and α a Probability).
TermSymbolDescription and notes1. Distribution function (df) or cumulative distribution function (cdf)F(x)F(x is the probability that the variate takes a value less than or equal to x. 2. Probability density function (pdf) (continuous variates)f(x)A function whose general integral over the range to is equal to the probability that the variate takes a value in that range. 3. Probability function (pf) (discrete variates)f(x)f(x) is the probability that the variate takes the value x. 4. Inverse distribution function or quantile function (of probability α) is the quantile such that the probability that the variate takes a value less than or equal to it is α. is the percentile. The relation to df and pdf is shown in Figures 2.3, 2.4, and 2.6.5. Survival functionS(x)S(x) is the probability that the variate takes a value greater than x. 6. Inverse survival function (of probability α) is the quantile that is exceeded by the variate with probability α. where S is the survival function and G is the inverse distribution function.7. Hazard function (or failure rate, hazard rate, or force of mortality)h(x)h(x) is the ratio of the probability density function to the survival function at quantile x. 8. Mills ratiom(x)m(x) is the inverse of the hazard function. 9. Cumulative or integrated hazard functionH(x)Integral of the hazard function. 10. Probability generating function (discrete nonnegative integer valued variates); also called the geometric or z transformP(t)A function of an auxiliary variable t (or alternatively z) such that the coefficient of .
