REDUCING TIME IN RELIABILITY STUDIES 199 



and if we compare it with the non-replacement case {g/Q is large) we 

 obtain 



^(n,) ^ -^. (i = 1, 2, • . • , n - 1). (47) 



These comparisons show that the difference in (46) is small when g/0 is 

 small and for j < n the difference in (47) is small when g/d is large. 



It is possible to compute E{Tnj) exactly for g ^ but the computa- 

 tion is extremely tedious for j ^ 2. The results for j = 1 and are given 

 in (44). Fori = 2 



E(Tn2) = 



n 



(n + 2)(/i - 1) -(n-2)gie 



1 - ' ' ': -e 



n 



+ Vl^iI g-(«-i)p/^ ri-2_ -un-i),ie I {n>2) 



n — \ v?{n — 1) 



and 



2{n-l)glB 



(48) 



E{T,.^ = ^ - ^ [1 - ^e-'" + e-'"'\ (49) 



For the case of two populations with a common guarantee period g 

 we can write similar inequalities. We shall use different symbols a, h for 

 the initial sample size from the populations with scale parameters Oi , O2 

 respectively even though our principal interest is in the case a = b = n 

 say. Let Ta,b.j denote the interval between the j^^ and j -f P* fail- 

 ures in this case and let X, = l/di (i = 1,2). We then have for all values 

 of a and b 



[aXi + b\o]-' ^ E(TaXj) ^ E(Taxo) 



= g + [aXi + b\,]-' (j = 0,1,2, ■■■, ^) (50) 



J?(T ^ (gl + g){e2 + g) .riN 



a{92 -h 9) + b{di + g) 



The result for E(Ta,b.i) corresponding to that in (43) does not hold if 

 the ratio di/62 is too large; in particular it can be shown that 



-0[(a-l)Xi+6X2l-l 



E{T.,b..) = ^ "^^ ^' ^ 



aXi + 6X2/ \(a — l)Xi 4- 6X2 



_ Xie 



aXi + 6X2 



+ / ^X2 Y 1 \r x^e-''^'^^''-''''-'- 



(52) 



,aXi -\- bX2/\aXi + (& — 1)^2 L 0X1 + ^^2 



is larger than E{Ta,h.J for a = 6 = 1 when ^/^i = 0.01 and g/di = 0.10 



