Introduction to Probability and Statistics for Engineers and Scientists

*7.6Confidence Interval of the Mean of the Exponential Distribution 265

*7.6Confidence Interval of the Mean of the Exponential Distribution

IfX 1 ,X 2 ,...,Xnare independent exponential random variables each having meanθ, then
it can be shown that the maximum likelihood estimator ofθis the sample mean

∑n
i= 1 Xi/n.
To obtain a confidence interval estimator ofθ, recall from Section 5.7 that

∑n
i= 1 Xihas
a gamma distribution with parametersn,1/θ. This in turn implies (from the relationship
between the gamma and chi-square distribution shown in Section 5.8.1.1) that

2

θ

∑n

i= 1

Xi∼χ 22 n

Hence, for anyα∈(0, 1)

P

{

χ 12 −α/2,2n<

2

θ

∑n

i= 1

Xi<χα^2 /2,2n

}

= 1 −α

or, equivalently,

P









2

∑n

i= 1

Xi

χα^2 /2,2n

<θ <

2

∑n

i= 1

Xi

χ 12 −α/2,2n









= 1 −α

Hence, a 100(1−α) percent confidence interval forθis

θ∈







2

∑n

i= 1

Xi

χα^2 /2,2n

,

2

∑n

i= 1

Xi

χ 12 −α/2,2n







EXAMPLE 7.6a The successive items produced by a certain manufacturer are assumed to
have useful lives that (in hours) are independent with a common density function

f(x)=

1

θ

e−x/θ,0<x<∞

If the sum of the lives of the first 10 items is equal to 1,740, what is a 95 percent confidence
interval for the population meanθ?

* Optional section.