Introduction to Probability and Statistics for Engineers and Scientists

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)