592 Chapter 14*:Life Testing
increases inθ(why?). Hence,
ifθ<θL, thenPθ{N(T)≤r}<PθL{N(T)≤r}=
α
2
ifθ>θU, thenPθ{N(T)≥r}<PθU{N(T)≥r}=
α
2
It remains to determineθLandθU. To do so, note first that the event thatN(T)≥ris
equivalent to the statement that therth failure occurs before or at timeT. That is,
N(T)≥r⇔X 1 +···+Xr≤T
and so
Pθ{N(T)≥r}=Pθ{X 1 +···+Xr≤T}
=P{γ(r,1/θ)≤T}
=P
{
θ
2
χ 22 r≤T
}
=P
{
χ 22 r≤ 2 T/θ
}
Hence, upon evaluating the foregoing atθ = θU, and using the fact thatP{χ 22 r ≤
χ 12 −α/2,2r}=α/2, we obtain that
α
2
=P
{
χ 22 r≤
2 T
θU
}
and that
2 T
θU
=χ 12 −α/2,2r
or
θU= 2 T/χ 12 −α/2,2r
Similarly, we can show that
θL= 2 T/χα^2 /2,2r
and thus the 100(1−α) percent confidence interval estimate forθis
θ∈(2T/χα^2 /2,2r,2T/χ 12 −α/2,2r)