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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Probability and Statistics
- Sprache: Englisch
Understand and utilize the latest developments in Weibull inferential methods While the Weibull distribution is widely used in science and engineering, most engineers do not have the necessary statistical training to implement the methodology effectively. Using the Weibull Distribution: Reliability, Modeling, and Inference fills a gap in the current literature on the topic, introducing a self-contained presentation of the probabilistic basis for the methodology while providing powerful techniques for extracting information from data. The author explains the use of the Weibull distribution and its statistical and probabilistic basis, providing a wealth of material that is not available in the current literature. The book begins by outlining the fundamental probability and statistical concepts that serve as a foundation for subsequent topics of coverage, including: * Optimum burn-in, age and block replacement, warranties and renewal theory * Exact inference in Weibull regression * Goodness of fit testing and distinguishing the Weibull from the lognormal * Inference for the Three Parameter Weibull Throughout the book, a wealth of real-world examples showcases the discussed topics and each chapter concludes with a set of exercises, allowing readers to test their understanding of the presented material. In addition, a related website features the author's own software for implementing the discussed analyses along with a set of modules written in Mathcad, and additional graphical interface software for performing simulations. With its numerous hands-on examples, exercises, and software applications, Using the Weibull Distribution is an excellent book for courses on quality control and reliability engineering at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for engineers, scientists, and business analysts who gather and interpret data that follows the Weibull distribution
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Table of Contents
Cover
WILEY SERIES IN PROBABILITY AND STATISTICS
Title page
Copyright page
Dedication
Preface
CHAPTER 1 Probability
1.1 SAMPLE SPACES AND EVENTS
1.2 MUTUALLY EXCLUSIVE EVENTS
1.3 VENN DIAGRAMS
1.4 UNIONS OF EVENTS AND JOINT PROBABILITY
1.5 CONDITIONAL PROBABILITY
1.6 INDEPENDENCE
1.7 PARTITIONS AND THE LAW OF TOTAL PROBABILITY
1.8 RELIABILITY
1.9 SERIES SYSTEMS
1.10 PARALLEL SYSTEMS
1.11 COMPLEX SYSTEMS
1.12 CROSSLINKED SYSTEMS
1.13 RELIABILITY IMPORTANCE
CHAPTER 2 Discrete and Continuous Random Variables
2.1 PROBABILITY DISTRIBUTIONS
2.2 FUNCTIONS OF A RANDOM VARIABLE
2.3 JOINTLY DISTRIBUTED DISCRETE RANDOM VARIABLES
2.4 CONDITIONAL EXPECTATION
2.5 THE BINOMIAL DISTRIBUTION
2.6 THE POISSON DISTRIBUTION
2.7 THE GEOMETRIC DISTRIBUTION
2.8 CONTINUOUS RANDOM VARIABLES
2.9 JOINTLY DISTRIBUTED CONTINUOUS RANDOM VARIABLES
2.10 SIMULATING SAMPLES FROM CONTINUOUS DISTRIBUTIONS
2.11 THE NORMAL DISTRIBUTION
2.12 DISTRIBUTION OF THE SAMPLE MEAN
2.13 THE LOGNORMAL DISTRIBUTION
2.14 SIMPLE LINEAR REGRESSION
CHAPTER 3 Properties of the Weibull Distribution
3.1 THE WEIBULL CUMULATIVE DISTRIBUTION FUNCTION (CDF), PERCENTILES, MOMENTS, AND HAZARD FUNCTION
3.2 THE MINIMA OF WEIBULL SAMPLES
3.3 TRANSFORMATIONS
3.4 THE CONDITIONAL WEIBULL DISTRIBUTION
3.5 QUANTILES FOR ORDER STATISTICS OF A WEIBULL SAMPLE
3.6 SIMULATING WEIBULL SAMPLES
CHAPTER 4 Weibull Probability Models
4.1 SYSTEM RELIABILITY
4.2 WEIBULL MIXTURES
4.3 P(Y < X)
4.4 RADIAL ERROR
4.5 PRO RATA WARRANTY
4.6 OPTIMUM AGE REPLACEMENT
4.7 RENEWAL THEORY
4.8 OPTIMUM BIDDING
4.9 OPTIMUM BURN-IN
4.10 SPARE PARTS PROVISIONING
CHAPTER 5 Estimation in Single Samples
5.1 POINT AND INTERVAL ESTIMATION
5.2 CENSORING
5.3 ESTIMATION METHODS
5.4 GRAPHICAL ESTIMATION OF WEIBULL PARAMETERS
5.5 MAXIMUM LIKELIHOOD ESTIMATION
5.6 ML ESTIMATION FOR THE WEIBULL DISTRIBUTION
CHAPTER 6 Sample Size Selection, Hypothesis Testing, and Goodness of Fit
6.1 PRECISION MEASURE FOR MAXIMUM LIKELIHOOD (ML) ESTIMATES
6.2 INTERVAL ESTIMATES FROM MENON’S METHOD OF ESTIMATION
6.3 HYPOTHESIS TESTING—SINGLE SAMPLES
6.4 OPERATING CHARACTERISTIC (OC) CURVES FOR ONE-SIDED TESTS OF THE WEIBULL SHAPE PARAMETER
6.5 OC CURVES FOR ONE-SIDED TESTS ON A WEIBULL PERCENTILE
6.6 GOODNESS OF FIT
6.7 LOGNORMAL VERSUS WEIBULL
CHAPTER 7 The Program Pivotal.exe
7.1 RELATIONSHIP AMONG QUANTILES
7.2 SERIES SYSTEMS
7.3 CONFIDENCE LIMITS ON RELIABILITY
7.4 USING Pivotal.exe FOR OC CURVE CALCULATIONS
7.5 PREDICTION INTERVALS
7.6 SUDDEN DEATH TESTS
7.7 DESIGN OF OPTIMAL SUDDEN DEATH TESTS
CHAPTER 8 Inference from Multiple Samples
8.1 MULTIPLE WEIBULL SAMPLES
8.2 TESTING THE HOMOGENEITY OF SHAPE PARAMETERS
8.3 ESTIMATING THE COMMON SHAPE PARAMETER
8.4 INTERVAL ESTIMATION OF A PERCENTILE
8.5 TESTING WHETHER THE SCALE PARAMETERS ARE EQUAL
8.6 MULTIPLE COMPARISON TESTS FOR DIFFERENCES IN SCALE PARAMETERS
8.7 AN ALTERNATIVE MULTIPLE COMPARISON TEST FOR PERCENTILES
8.8 THE PROGRAM MULTI-WEIBULL.EXE
8.9 INFERENCE ON P (Y < X)
CHAPTER 9 Weibull Regression
9.1 THE POWER LAW MODEL
9.2 ML ESTIMATION
9.3 EXAMPLE
9.4 PIVOTAL FUNCTIONS
9.5 CONFIDENCE INTERVALS
9.6 TESTING THE POWER LAW MODEL
9.7 MONTE CARLO RESULTS
9.8 EXAMPLE CONCLUDED
9.9 APPROXIMATING u* AT OTHER STRESS LEVELS
9.10 PRECISION
9.11 STRESS LEVELS IN DIFFERENT PROPORTIONS THAN TABULATED
9.12 DISCUSSION
9.13 THE DISK OPERATING SYSTEM (DOS) PROGRAM REGEST
CHAPTER 10 The Three-Parameter Weibull Distribution
10.1 THE MODEL
10.2 ESTIMATION AND INFERENCE FOR THE WEIBULL LOCATION PARAMETER
10.3 TESTING THE TWO- VERSUS THREE-PARAMETER WEIBULL DISTRIBUTION
10.4 POWER OF THE TEST
10.5 INTERVAL ESTIMATION
10.6 INPUT AND OUTPUT SCREENS OF LOCEST.exe
10.7 THE PROGRAM LocationPivotal.exe
10.8 SIMULATED EXAMPLE
CHAPTER 11 Factorial Experiments with Weibull Response
11.1 INTRODUCTION
11.2 THE MULTIPLICATIVE MODEL
11.3 DATA
11.4 ESTIMATION
11.5 TEST FOR THE APPROPRIATE MODEL
11.6 MONTE CARLO RESULTS
11.7 THE DOS PROGRAM TWOWAY
11.8 ILLUSTRATION OF THE INFLUENCE OF FACTOR EFFECTS ON THE SHAPE PARAMETER ESTIMATES
11.9 NUMERICAL EXAMPLES
Index
WILEY SERIES IN PROBABILITY AND STATISTICS
The Wiley Series in Probability and Statistics is well established and authoritative. It covers many topics of current research interest in both pure and applied statistics and probability theory. Written by leading statisticians and institutions, the titles span both state-of-the-art developments in the field and classical methods.
Reflecting the wide range of current research in statistics, the series encompasses applied, methodological and theoretical statistics, ranging from applications and new techniques made possible by advances in computerized practice to rigorous treatment of theoretical approaches.
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Notes
*Now available in a lower priced paperback edition in the Wiley Classics Library.
†Now available in a lower priced paperback edition in the Wiley–Interscience Paperback Series.
Cover image: Courtesy of the author.
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging-in-Publication Data:
McCool, John, 1936–
Using the Weibull distribution : reliability, modeling, and inference / John I. McCool.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-118-21798-6 (cloth)
1. Weibull distribution—Textbooks. 2. Probabilities—Textbooks. I. Title.
QA273.6.M38 2012
519.2'4–dc23
2012002909
To Jim, Fran, and PegandTo Mary, John, and Jen
Preface
This book grew out of my experience as a young mechanical engineer working in the research organization of the U.S. subsidiary of SKF, an international ball and roller bearing manufacturer. The ball and roller bearing industry adopted the use of the Weibull distribution to describe the fatigue life of its products back in the 1940s and before the appearance of Weibull’s influential article in 1951 that set the stage for its present enormous popularity (Weibull, 1951). I have tried to write the book that I would wish to have if I were today a young engineer asked to become my company’s “Weibull Guy.”
I began to organize the material in order to teach a number of short courses sponsored by the American Society of Mechanical Engineers (ASME), the U.S. Navy, and later by the American Bearing Manufacturers Association (ABMA). The book benefited from my experience as an adjunct and, since 1988, as a full-time Professor of Systems Engineering at Penn State’s School of Graduate Professional Studies in Malvern, Pennsylvania, where I have taught master’s level courses in statistics, quality control, and reliability engineering, among others. I have twice used a draft of the book as the text for the Reliability Engineering course in the Master of Systems Engineering curriculum. A sabbatical provided the opportunity to put much of the material into its present form.
The book has also benefited immensely from the opportunity I had while with SKF to develop techniques for inference on the Weibull distribution under the sponsorship of the Aerospace Research Laboratory at Wright Patterson Air Force Base and the Air Force Office of Scientific Research.
The availability of the digital computer is responsible for much of the progress in the development of tools for inference for the Weibull distribution that has taken place since the pioneering work of Lieblein and Zelen (1956). They used a computer to determine the variances and covariances of order statistics needed in the construction of best linear unbiased estimates of the Weibull parameters. Later the computer was essential for making practical the conduct of Monte Carlo determinations of quantities needed for inference and for the computation of estimates for individual samples or sets of samples using the method of maximum likelihood.
The astonishing evolution of personal computing now makes sophisticated inferential techniques accessible from the desktop of the engineer or scientist. Although available computing power is more than equal to the task, many powerful techniques for extracting the most information from expensive data have not yet been as widely adopted as they should because appropriate software has been unavailable. It is my hope that this book and the software that accompanies it will help rectify that situation.
The use of software developed by a succession of talented programmers who have helped me over the years is illustrated in this book. The software may be downloaded for free from my website http://www.personal.psu.edu/mpt. It includes a set of disk operating system (DOS) programs that should serve until such a time as these capabilities become incorporated into commercial software packages. There is also a set of modules written in Mathcad, which both document and perform the calculations associated with parameter estimation in various settings and with other quantitative models such as optimum age replacement and optimum burn-in. Finally it includes some graphical interface software that performs simulations needed for inference in single and multiple samples from the two-parameter Weibull, inference on the location parameter of the three-parameter Weibull, and for determining critical values for a test of goodness of fit.
The reader will find that the exposition is liberally accompanied by numerical examples. All of the chapters conclude with a set of exercises that may be used to test the reader’s mastery of the material or as class exercises in a short course.
The Weibull distribution cannot be discussed in isolation from the rest of statistical methodology. A knowledge of elementary probability theory and of distributions such as the binomial, Poisson, exponential, normal, and lognormal is essential for its effective application. The book is intended to be self-contained in this regard, with the inclusion of two chapters treating the fundamentals of probability and statistics needed for what follows.
Chapter 1 introduces the ideas of mutual exclusivity, independence, conditional probability, and the Law of Total probability as they impact reliability-related calculations. The chapter concludes with the application of probability principles to the computation of the reliability of systems in terms of the reliability of its components. Combinations of series, parallel, and crosslinked systems are considered. The idea of the reliability importance of a component is introduced.
Chapter 2 describes singly and jointly distributed discrete and continuous random variables and the concepts of the mean, variance, and covariance. The binomial, Poisson, and geometric discrete random variables are introduced. The binomial distribution is applied to the task of computing the reliability of a system that comprises n identical components and functions as long as at least k (≤n) of the components function successfully. The uniform, normal, and lognormal continuous random variables are discussed and their use is illustrated with examples. The hazard function is introduced and its relationship to the reliability and density functions is explained. Finally the use of the inverse transformation method for simulating samples from a continuous distribution is described and illustrated by an example.
Chapter 3 enumerates the properties of the Weibull distribution. It shows the Weibull to be a generalization of the exponential distribution. It gives the expressions for the mean, variance, mode, skewness, hazard function, and quantiles of the Weibull distribution in terms of its two parameters. The logarithmic transform of a Weibull random variable is shown to follow the distribution of smallest extremes. The power transformation of a Weibull random variable is shown to map it into a different member of the Weibull family. The Weibull distribution conditional on exceeding a specified value is derived and shown to apply to the interpretation of the practice of burn-in used to improve the life of electronic equipment. The computation of the mean residual life is discussed as is the simulation of samples from a Weibull population.
Chapter 4 describes a number of useful applications of the two-parameter Weibull distribution valid when its parameters are known or assumed. This includes the distribution of mixtures of Weibull random variables, the computation of P(Y < X) when X and Y are both Weibull distributed with a common shape parameter, the Weibull distribution of radial error when the location errors in two orthogonal directions are independent and are normally distributed with a common variance. Three types of warranties are discussed (i) a pro rata warranty, (ii) a free replacement warranty, and (iii) a renewing free replacement warranty. Two preventive maintenance strategies are considered: (i) age replacement and (ii) block replacement. Also discussed are optimum bidding in a sealed bid competition and spare parts provisioning when the life distribution is exponential. Weibull renewal theory is discussed and applied to the analysis of the block replacement warranty.
Chapter 5 treats estimation in single samples. The notions of bias and precision of estimation are illustrated by means of the sampling distributions of two competing estimators of the 10th percentile of a Weibull population. Graphical estimation, which for years was the standard means of estimating the Weibull parameters, is explained and various choices of plotting positions are described and compared. Hazard plotting and the Kaplan–Meier method for graphical plotting of randomly censored data are explained. The method of maximum likelihood is introduced and applied to the exponential special case of the Weibull. Complete and type II censoring is discussed. Maximum likelihood estimation for the exponential is applied for the Weibull when the shape parameter is known. Software is illustrated for maximum likelihood estimation of the Weibull parameters in complete and type II censored samples when both parameters are unknown. Exact interval estimation of the shape parameter and percentiles is considered for complete and type II censored samples. Asymptotic results generally applied in the type I censored case are explained and a Mathcad module is given for performing those calculations.
Chapter 6 deals with sample size selection, as well as hypothesis testing. The question most frequently asked by experimenters of their statistical gurus is “what sample size do I need?” Two approaches are offered in this chapter both of which have their parallels in normal distribution theory. One approach is to display for each sample size a precision measure that reflects the tightness of confidence limits on the shape parameter and/or a percentile. The experimenter must express a goal or target value for either of these precision measures, and the needed sample size is determined by a table lookup. The second approach is in terms of the operating characteristic function for a hypothesis test. The experimenter specifies a target value of the shape parameter or a percentile and the desired large probability that the sample will be accepted if the population value equals that target value. The experimenter also must specify an undesirable value of the percentile or shape parameter and the desired small probability that a population having that value will be accepted. Both approaches rely on percentage points of certain pivotal quantities only sparsely available in the current literature. Values are available for just a few of the Weibull percentiles of interest and for limited combinations of sample sizes and numbers of failures. Software described fully in Chapter 7 performs the simulations needed to remove these limitations.
Chapter 6 concludes with a discussion of how to test the hypothesis that a data sample was drawn from a two-parameter Weibull distribution. The discussion distinguishes between the completely specified case in which the Weibull parameters are specified and the more usual case in which the parameters are estimated by the method of maximum likelihood. The computations for the Kolmogorov–Smirnov test are illustrated for an uncensored sample, but the discussion mainly focuses on the Anderson–Darling (AD) test, which has been shown in many studies to be among the more powerful methods of testing goodness of fit to the Weibull. Critical values of the AD statistic may be found in the literature but are not available for many combinations of sample sizes and censoring amounts. A simulation program called ADStat is introduced to overcome this problem. It computes 19 percentage points of the distribution of the AD statistic. It handles complete and type II censored data and will accommodate both the case where the Weibull parameters are completely specified and the case where they need to be estimated. The software will also compute the AD statistic for the user’s data sample if needed.
The Weibull and lognormal distribution often compete as models for life test data. Two methods are described and illustrated by examples for testing whether an uncensored sample follows the lognormal or the two-parameter Weibull distribution.
Chapter 7 is devoted to an exposition of the features of the simulation software program Pivotal.exe. Exact inference, that is, confidence limits and hypothesis tests for the Weibull distribution parameters and percentiles based on maximum likelihood estimation in complete or type II censored samples, requires the determination via simulation of percentage points of the distribution of certain pivotal quantities. The program Pivotal.exe described in this chapter performs these calculations for user-selected sample sizes and percentiles. The software is also applicable to inference on series systems of identical Weibull-distributed components and for analyzing the results of sudden death tests.
The software allows the output of 10,000 paired values of the ML estimates of the Weibull shape and scale parameters, which can be post-processed using spreadsheet-based software to provide: (i) operating characteristic curves for hypothesis tests on the Weibull shape parameter or a percentile (ii) confidence intervals on reliability at a specified life and (iii) prediction intervals for a future value.
Chapter 8 is concerned with inference from multiple samples. It discusses how to make best use of the data that result when, not uncommonly, a set of tests is performed differing with respect to the level of some factor such as a design feature, a material, or a lubricant type. Provided that it can be assumed that the shape parameter is the same among the sampled populations, data may be pooled to provide tighter confidence limits for the common shape parameter and for the percentiles of interest among the individual populations. A hypothesis test is provided for assessing whether the common shape parameter assumption is tenable and a simulation software program is described for generating the critical values needed for conducting the test among k sets of samples of size n which are complete or censored at the r-th order statistic. A test for the equality of the scale parameters among the tests is also given which is analogous to the one-way analysis of variance in normal distribution theory as is a multiple comparison test for differences among the scale parameters. Too often in published work reporting on sets of tests, the analysis goes no further than the subjective assessment of a set of Weibull plots.
The chapter also contains a method for setting confidence limits for the value of P(X < Y) when X and Y are Weibull distributed with a common shape parameter based on random samples drawn from the two distributions. P(X < Y) is considered the reliability when X represents a random stress and Y a random strength.
Chapter 9, Weibull Regression, refers to the situation where testing is conducted at several levels of a quantitative factor that might be a stress or load. The Weibull scale parameter is assumed to vary as a power function of the level to which this factor is set, while the shape parameter is assumed not to vary with the level of the factor. Such a model finds use in the analysis of accelerated tests wherein the purpose is to complete testing quickly by running the tests at stresses higher than encountered under use conditions. The fitted model is then used to extrapolate to the lower stress values representative of “use” conditions. The factor need not necessarily be one that degrades performance. The model can account for the scale parameter either increasing or decreasing as a power of the factor level. It is shown that exact confidence intervals may be computed for the power function exponent, the shape parameter, and a percentile of the distribution at any level of the factor. Tables of the critical values are given for two and three levels of the factor. The chapter includes a description of a DOS program for performing the calculations to analyze a set of data, along with a Mathcad module that illustrates the solution of the relevant equations.
The three-parameter Weibull distribution is covered in Chapter 10. It generalizes the two-parameter Weibull to include the location or threshold parameter representing an amount by which the probability density function is offset to the right. In the context of life testing, this offset represents a guarantee time prior to which failure cannot occur. Tables are given for a range of sample sizes, whereby one may (i) test whether the location parameter is zero and (ii) determine a lower confidence limit for the location parameter. A DOS program for performing the computations on a data set is described and a simulation program is given for extending the tabled values to other sample sizes and for exploring the power of the test.
Chapter 11 is entitled Factorial Experiments with Weibull Response. Tests conducted at all combinations of the levels of two or more factors are called factorial experiments. Factorial experiments have been shown to be more efficient in exploring the effects of external factors on a response variable than nonfactorial arrangements of factor levels. In this chapter we present a methodology for the analysis of Weibull-distributed data obtained at all combinations of the levels of two factors. Item life is assumed to follow the two-parameter Weibull distribution with a shape parameter that, although unknown, does not vary with the factor levels. The purpose of the analysis is (i) to compute interval estimates of the common shape parameter and (ii) to assess whether either factor has a multiplicative effect on the Weibull scale parameter and hence on any percentile of the distribution. A DOS program is included for performing the analysis. Tables are given for hypothesis testing for various uncensored sample sizes for the 2 × 2, 2 × 3, and 3 × 3 designs.
I am grateful for support for my work by H.L. Harter, formerly of the Air Force Aerospace Research Laboratory at Wright Patterson Air Force Base, and by the late I. Shimi of the Air Force Office of Scientific Research. I am grateful as well for the help of a number of people in developing the software and tables included in this volume. Mr. John C. Shoemaker, a colleague at SKF, wrote the software used for generating the tables of the distribution of pivotal quantities originally published as an Air Force report. Mr. Ted Staub, when a Penn State student, developed an early version of the Pivotal.exe software, which generates the distribution of pivotal quantities for user-selected sample sizes. Another student, Donny Leung, provided the user interface for Pivotal.exe and extended it produce the Multi-Weibull software. A colleague, Pam Vercellone, helped refine Pivotal.exe and Multi-Weibull. Student Nimit Mehta developed the ADStat software described in Chapter 6. Student Christopher Garrell developed the LocationPivotal software described in Chapter 10.
Finally, I would like to acknowledge the wishes of my grandchildren Jaqueline and Jake to see their names in a book.
JOHN I. MCCOOL
REFERENCES
Lieblein, J. and M. Zelen. 1956. Statistical investigation of the fatigue life of deep-groove ball bearings. Journal of Research of the National Bureau of Standards 57(5): 273–316.
Weibull, W. 1951. A statistical distribution function of wide applicability. Journal of Applied Mechanics 18(3): 293–297.
CHAPTER 1
Probability
The study of reliability engineering requires an understanding of the fundamentals of probability theory. In this chapter these fundamentals are described and illustrated by examples. They are applied in Sections 1.8 to 1.13 to the computation of the reliability of variously configured systems in terms of the reliability of the system’s components and the way in which the components are arranged.
Probability is a numerical measure that expresses, as a number between 0 and 1, the degree of certainty that a specific outcome will occur when some random experiment is conducted. The term random experiment refers to any act whose outcome cannot be predicted. Coin and die tossing are examples. A probability of 0 is taken to mean that the outcome will never occur. A probability of 1.0 means that the outcome is certain to occur. The relative frequency interpretation is that the probability is the limit as the number of trials N grows large, of the ratio of the number of times that the outcome of interest occurs divided by the number of trials, that is,
(1.1)
where n denotes the number of times that the event in question occurs. As will be seen, it is sometimes possible to deduce p by making assumptions about the relative likelihood of all of the other events that could occur. Often this is not possible, however, and an experimental determination must be made. Since it is impossible to conduct an infinite number of trials, the probability determined from a finite value of N, however large, is considered an estimate of p and is distinguished from the unknown true value by an overstrike, most usually a caret, that is, .
1.1 SAMPLE SPACES AND EVENTS
The relationship among probabilities is generally discussed in the language of set theory. The set of outcomes that can possibly occur when the random experiment is conducted is termed the sample space. This set is often referred to by the symbol Ω. As an example, when a single die is tossed with the intent of observing the number of spots on the upward face, the sample space consists of the set of numbers from 1 to 6. This may be noted symbolically as Ω = {1, 2, … 6}. When a card is drawn from a bridge deck for the purpose of determining its suit, the sample space may be written: Ω = {diamond, heart, club, spade}. On the other hand, if the purpose of the experiment is to determine the value and suit of the card, the sample space will contain the 52 possible combinations of value and suit. The detail needed in a sample space description thus depends on the purpose of the experiment. When a coin is flipped and the upward face is identified, the sample space is Ω = {Head, Tail}. At a more practical level, when a commercial product is put into service and observed for a fixed amount of time such as a predefined mission time or a warranty period, and its functioning state is assessed at the end of that period, the sample space is Ω = {functioning, not functioning} or more succinctly, Ω = {S, F} for success and failure. This sample space could also be made more elaborate if it were necessary to distinguish among failure modes or to describe levels of partial failure.
Various outcomes of interest associated with the experiment are called Events and are subsets of the sample space. For example, in the die tossing experiment, if we agree that an event named A occurs when the number on the upward face of a tossed die is a 1 or a 6, then the corresponding subset is A = {1, 6}. The individual members of the sample space are known as elementary events. If the event B is defined by the phrase “an even number is tossed,” then the set B is {2, 4, 6}. In the card example, an event C defined by “card suit is red” would define the subset C = {diamond, heart}. Notationally, the probability that some event “E” occurs is denoted P(E). Since the sample space comprises all of the possible elementary outcomes, one must have P(Ω) = 1.0.
1.2 MUTUALLY EXCLUSIVE EVENTS
Two events are mutually exclusive if they do not have any elementary events in common. For example, in the die tossing case, the events A = {1, 2} and B = {3, 4} are mutually exclusive. If the event A occurred, it implies that the event B did not. On the other hand, the same event A and the event C = {2, 3, 4} are not mutually exclusive since, if the upward face turned out to be a 2, both A and C will have occurred. The elementary event “2” belongs to the intersection of sets A and C. The set formed by the intersection of sets A and C is written as A∩C. The probability that the outcome will be a member of sets A and C is written as P(A∩C).
When events are mutually exclusive, the probabilities associated with the events are additive. One can then claim that the probability of the mutually exclusive sets A and B is the sum of P(A) and P(B).
In the notation of set theory, the set that contains the elements of both A and B is called the union of A and B and designated A ∪ B. Thus, one may compute the probability that either of the mutually exclusive events A or B occurs as:
(1.2)
The same result holds for three or more mutually exclusive events; the probability of the union is the sum of the probabilities of the individual events.
The elementary events of a sample space are mutually exclusive, so for the die example one must have:
(1.3)
Now reasoning from the uniformity of shape of the die and homogeneity of the die material, one might make a leap of faith and conclude that the probability of the elementary events must all be equal and so,
If that is true then the sum in Equation 1.3 will equal 6p, and, since 6p = 1, p = 1/6. The same kind of reasoning with respect to coin tossing leads to the conclusion that the probability of a head is the same as the probability of a tail so that P(H) = P(T) = 1/2. Dice and coins whose outcomes are equally likely are said to be “fair.” In the card selection experiment, if we assume that the card is randomly selected, by which we mean each of the 52 cards has an equal chance of being the one selected, then the probability of selecting a specific card is 1/52. Since there are 13 cards in each suit, the probability of the event “card is a diamond” is 13/52 = 1/4.
1.3 VENN DIAGRAMS
Event probabilities and their relationship are most commonly displayed by means of a Venn diagram named for the British philosopher and mathematician John Venn, who introduced the Venn diagram in 1881. In the Venn diagram a rectangle symbolically represents the set of outcomes constituting the sample space Ω; that is, it contains all of the elementary events. Other events, comprising subsets of the elementary outcomes, are shown as circles within the rectangle. The Venn diagram in Figure 1.1 shows a single event A.
Figure 1.1 Venn diagram showing a single event A.
The region outside of the circle representing the event contains all of the elementary outcomes not encompassed by A. The set outside of A, Ω-A, is generally called “not-A” and is indicated by a bar overstrike . Since A and are mutually exclusive and sum to the whole sample space, we have:
(1.4)
Therefore, the probability of the event not-A may be found simply as:
(1.5)
Thus, if A is the event that a bearing fails within the next 1000 hours, and P(A) = 0.2, the probability that it will survive is 1 − 0.2 = 0.8. The odds of an event occurring is the ratio of the probability that the event occurs to the probability that it does not. The odds that the bearing survives are thus 0.8/0.2 = 4 or 4 to 1.
Since mutually exclusive events have no elements in common, they appear as nonoverlapping circles on a Venn diagram as shown in Figure 1.2 for the two mutually exclusive events A and B:
Figure 1.2 Venn diagram for mutually exclusive events A and B.
1.4 UNIONS OF EVENTS AND JOINT PROBABILITY
The Venn diagram in Figure 1.3 shows two nonmutually exclusive events, A and B, depicted by overlapping circles. The region of overlap represents the set of elementary events shared by events A and B. The probability associated with the region of overlap is sometimes called the joint probability of the two events.
Figure 1.3 Venn diagram for overlapping events A and B.
In this case, computing the probability of the occurrence of event A or B or both as the sum of P(A) and P(B) will add the probability of the shared events twice. The correct formula is obtained by subtracting the probability of the intersection from the sum of the probabilities to correct for the double inclusion:
(1.6)
As an example, consider again the toss of a single die with the assumption that the elementary events are equally likely and thus each have a probability of occurrence of p = 1/6. Define the events A = {1, 2, 3, 4} and B = {3, 4, 5}. Then P(A) = 4/6, P(B) = 3/6 and since the set (A ∩ B) = {3, 4}, it follows that P(A ∩ B) = 2/6. The probability that the event A or B occurs may now be written:
The formula above applies even to mutually exclusive events when it is recalled that for mutually exclusive events,
Similar reasoning leads to the following expression for the union of three events:
(1.7)
