Table Of ContentSTATISTICAL
ANALYSIS
OF RELIABILITY
AND LIFE-TESTING
MODELS
STATISTICAL
ANALYSIS
OF RELIABILITY
AND LIFE-TESTING
MODELS
Theory and Methods
Second Edition
LEE J. BAIN
University of Missouri-Rolla
Rolla, Missouri
MAX ENGELHARDT
University of Missouri-Rolla
Rolla, Missouri
and
Idaho National Engineering laboratory
EG&G Idaho, Inc.
Idaho Falls, Idaho
Marcel Dekker, Inc. New York • Basel • Hong Kong
Library of Congress Cataloging-i~-Publication Data
Bain, Lee J .
Statistical analysis of reliability and life-testing models:
theory and methods/Lee J. Bain, Max Engelhardt. -- 2nd ed.
p. cm. -- (Statistics, textbooks and monographs; vol. 115)
Includes bibliographical references (p. ) and index.
ISB 0-8247-8506-1
1. Reliability (Engineering)--Statistical methods. 2. Accelerated
life testing. 3. Distribution (Probability theory) 4. Mathematical
statistics. I. Engelhardt, Max. II. Title. III. Series:
Statistics, textbooks and monographs; v. 115.
TS173.B34 1991
620' .00452--dc20 91-7832
CIP
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Copyright © 1991 by MARCEL DEKKER, I C. All Rights Reserved
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1
Probabilistic Models
1. PROBABILITY
1.1. Introduction
Suppose a certain physical phenomenon is of interest, and an experi
ment is conducted to obtain an observed value of this phenomenon.
It may be possible to develop a mathematical model so that if the
conditions of the experiment are known, then the outcome can he
predetermined, at least to a sufficient degree of accuracy. For
example, Ohm's law, E = IR, predicts the value of the electromotive
force for a given level of current and resistance. Ohm's law is an
example of a deterministic mathematical model. However, it may be
that the outcome of the experiment cannot be determined on the ba-·
sis of the available knowledge of the experiment. In this case a
probabilistic mathematical model is necessary.
111ere are many different reasons why a probabilistic model may
be required. In some cases the outcomes may truly occur by chance,
such as when a die is rolled, any of the six possible faces may oc
cur, on the other hand, it may be that some of the conditions of
the experiment are simply unknown or cannot be controlled. For ex
ample, two light bulbs may be manufactured by the same process and
used under the same general conditions but still fail at different
times. Similarly, a deterministic model may exist but be too com-
1
2 Chapter 1
plex to develop, or some factor such as measurement error may nv
cessitate the use of a probabilistic model.
The purpose of this chapter is to set up the mathematical
structure for describing a physical phenomenon in terms of a prob-
ability model. Chapter 2 will review some of the general statis
tical procedures which are helpful in analyzing and selecting an
appropriate probability distribution to fit a p;_irticular prohlL'm
of interest. Specific statistical techniques are then studied in
detail in the following chapters for the most common probability
models in use.
1.2. Notation
Again suppose some physical phenomenon is of interest, such as
some characteristic of an item in a large population. This popu
lation may be either real or conceptual. For example, if the char
acteristic of interest is the length of time a light bulb made by
a certain process will function properly, then there may be a large
number of light bulbs already made, or, conceptually, there could
be a large number made. Of course, the goal is to gain knowledge
about this conceptual population of lifetimes of light bulbs made
under this process, based on a limited number of observed failure
times. The knowledge inferred about the population could then be
used in making decisions.
The process of obtaining an observed value of the physical
phenomenon is called an experiment. An observed value is called
an outcome, and the set of all possible outcomes of the experiment,
only one of which will occur on any given trial, is called the
sample space. It is mathematically convenient to associate the
outcomes with real numbers. A random variable X is a real-valued
function defined on a sample space which associates a real number,
x(e), with each possible outcome e. That is, saying that the par
ticular outcome e occurred is equivalent to saying that the exper-
Probabilistic Models 3
iment resulted in an observed value, X = x(e), of the random vari
able X. The set of possible values which the variable X may assume
is called the range X of X.
Example 1.2.l Suppose a coin is flipped once, and either
a head (H) or tail (T) may occur. Consider a random variable X
which associates the real number with a head, x(H) = 1, and the
number 0 with a tail, x(T) = 0. In this case the range of X is
only the two values 0 and 1. Any situation with only two possible
outcomes, such as success-failure or defective-nondefective items,
would be mathematically equivalent to the coin-tossing example.
Now, the probability that a head occurs may be expressed as
the probability that the random variable assumes a value 1,
Pr'[X = l]. Suppose that the probability that a head occurs is p,
and the probability a tail occurs is 1 - p. A value of p = 1/2,
for example, would be reasonable if the coin is well balanced.
Now this situation may be easily expressed mathematically by the
model
f (x) = p x (1 - p) 1-x , x = 0, 1 (1)
That is, Pr[H occurs] = Pr[X l] = f(l) = p, and Pr[T occurs]
Pr[X = O] = f(O) = 1 - p. Clearly the above random variable was a
convenient way of associating the outcomes with real numbers, how
ever some other random variable could have been defined if de
sired.
Example 1.2.2 Suppose a coin is flipped until a head oc-
curs, and suppose the characteristic of interest is the number of
flips required. In this case it is natural to simply define a ran
dom variable X to be the number of flips required to produce a
head. The range of X irt this case is the set of positive integers
X = {l, 2, 3, ... }. A possible probability model for this phe
nomenon could be denoted by
Pr [ X = x] = f(x) = (1 - p) x-1 p, x = 1, 2, 3, ...
where pis some positive constant between 0 and 1, that is,
4 Chapter 1
Prfl flip required]= f(l) = p
Prf2 flips required]= f(2) = (1 - p)p
Prfn flips require<l] = f(n) = (1 - p)11- 1p
Note that the above probability model could also apply to
many other variables, such as the number of draws required to ob
tain a defective light bulb.
The number of positive integers is said to be "countably in
finite." A sample space is called discrete if there is only a fi
nite or countably infinite number of possible outcomes.
Example 1. 2. 3 Suppose the random variable X denotes the
length of time a certain type of U ght bulb wi 11 fun ct ion proper I)'
before failure. In this case the range of X may be considered to
be the positive real line (0 ~ x < 00); however, it may sometimes
be convenient in such cases to refer to the whole real line
(-00 < x < 00) as the range of the variable, even though a negative
value would never actually occur. It shold also be noted that
more than one variable could be defined relative to the same ex
periment. In this example another random variable, say Y, could
denote the candlepower of the selected light bulb. Either of the
variables could be considered separately, or the outcome of the
experiment could be considered to be the ordered pair (x, y), con
stituting an observed value of the random vector (X, Y) with range
in two-dimensional Euclidean space.
If the range of a random variable is the real line, or more
generally Euclidean n space for a random vector, then the sample
space is said to be continuous. A sample space may also be mixed,
partly discrete and partly continuous. An event is an appropriate
collection of possible outcomes, i.e., a subset of the sample
space. Thus an event A is said to occur if the result of the ex-
periment is any one of the elements in the subset A . If A and B
denote two events, then A u B (A union B) will denote the set of
sample points or outcomes which are in either A or B (or both).
Probabilistic Models 5
That is, the event A u B occurs if either A or B occurs. The in
tersection of two sets A n B is the set of outcomes which are in
both A and B. If A and B have no points in common, they are said
to be mutually exclusive (disjoint). That is, A and Bare dis
joint if A n B = 0, where 0 denotes the empty set and A and B can
not both occur in this case. The complement of A, A, will denote
the event that A does not occur. Clearly either A or A will occur
but not both, so that A n A = 0 and A u A = S, where S denotes the
whole sample space.
An elementary event will refer to a set consisting of a
single outcome or point in a sample space. In a discrete sample
space any subset A may be considered to be an event, and A may be
expressed as A= .u00 E., where E. are disjoint elementary events.
1= 1 1 1
(Technically, any "Borel set" may be considered an event in a con-
tinuous sample space; however, any event in the above form is a
Borel set and any Borel set of practical importance can be ex
pressed in the above form.)
Note also that for any event expressed in the original sample
space, there will be a corresponding subset or event in the range
space of the associated random variable. For convenience, events
will usually be expressed directly in terms of subsets of the
range space of the random variable rather than as subsets of the
original sample space. In Ex. 1.2.2 the event A= {1,2,3,4}
{xjx = 1,2,3,4} would be the event of getting a head in four or
less flips. Thus, if one trial of the experiment is conducted,
the event A will have occurred if the observed value of X, say x,
is any integer from 1 to 4.
1.3. Definition of Probability
Although a general knowledge of probability was assumed in dis
cussing the earlier examples, it is necessary to state a more ex-
