Introduction to Probability and Statistics for Engineers and Scientists

14.3The Exponential Distribution in Life Testing 597

foregoing we see that

f(data|λ)=Kλre−λt

If we suppose prior to testing, thatλis distributed according to the prior densityg(λ),
then the posterior density ofλgiven the observed data is as follows:

f(λ|data)=

f(data|λ)g(λ)

∫

f(data|λ)g(λ)dλ

=

λre−λtg(λ)

∫

λre−λtg(λ)dλ

(14.3.11)

The preceding posterior density becomes particularly convenient to work with whengis
a gamma density function with parameters, say, (b,a) — that is, when

g(λ)=

ae−aλ(aλ)b−^1

(b)

, λ> 0

for some nonnegative constantsaandb. Indeed for this choice ofg we have from
Equation 14.3.11 that

f(λ|data)=Ce−(a+t)λλr+b−^1

=Ke−(a+t)λ[(a+t)λ]b+r−^1

whereCandKdo not depend onλ. Because we recognize the preceding as the gamma
density with parameters (b+r,a+t), we can rewrite it as

f(λ|data)=

(a+t)e−(a+t)λ[(a+t)λ]b+r−^1

(b+r)

, λ> 0

In other words, if the prior distribution ofλis gamma with parameters (b,a), then no
matter what the testing scheme, the (posterior) conditional distribution ofλgiven the
data is gamma with parameters (b+R,a+τ), whereτandRrepresent respectively the
total-time-on-test statistic and the number of observed failures. Because the mean of a
gamma random variable with parameters (b,a) is equal tob/a(see Section 5.7), we can
conclude thatE[λ|data], the Bayes estimator ofλ,is

E[λ|data]=

b+R

a+τ

EXAMPLE 14.3e Suppose that 20 items having an exponential life distribution with an
unknown rateλare put on life test at various times. When the test is ended, there have
been 10 observed failures — their lifetimes being (in hours) 5, 7, 6.2, 8.1, 7.9, 15, 18,