Introduction to Probability and Statistics for Engineers and Scientists

14.3The Exponential Distribution in Life Testing 585

and so, by independence, the joint probability density ofXij,j=1,...,ris

fXi 1 ,...,Xir(x 1 ,...,xr)=

∏r

j= 1

1

θ

e−xj/θ

Also, the probability that the othern−rof theX’s are all greater thanxris, again using
independence,

P{Xj>xrforj=i 1 ori 2 ...orir}=(e−xr/θ)n−r

Hence, we see that the likelihood of the observed data — call it L(x 1 ,...,xr,
i 1 ,...,ir) — is, forx 1 ≤x 2 ≤ ··· ≤xr,

L(x 1 ,...,xr,i 1 ,...,ir) (14.3.2)

=fXi 1 ,Xi 2 ,...Xir(x 1 ,...,xr)P{Xj>xr,j=i 1 ,...,ir}

=

1

θ

e−x^1 /θ···

1

θ

e−xr/θ(e−xr/θ)n−r

=

1

θr

exp









−

∑r

i= 1

xi

θ

−

(n−r)xr

θ









REMARK

The likelihood in Equation 14.3.2 not only specifies that the firstrfailures occur at
timesx 1 ≤x 2 ≤ ··· ≤xrbut also that theritems to fail were, in order,i 1 ,i 2 ,...,ir.
If we only desired the density function of the firstrfailure times, then since there are
n(n−1)···(n−(r−1))=n!/(n−r)!possible (ordered) choices of the firstritems to
fail, it follows that the joint density is, forx 1 ≤x 2 ≤ ··· ≤xr,

f(x 1 ,x 2 ,...,xr)=

n!

(n−r)!

1

θr

exp











−

∑r

i= 1

xi

θ

−

(n−r)

θ

xr











To obtain the maximum likelihood estimator ofθ, we take the logarithm of both sides
of Equation 14.3.2. This yields

logL(x 1 ,...,xr,i 1 ,...,ir)=−rlogθ−

∑r

i= 1

xi

θ

−

(n−r)xr

θ