REDUCING TIME IN RELIABILITY STUDIES 201 



upper bound, which holds for all \'aliies of c is given by 



E{T) ^ E{F- b)E{Tn,n..) = E{F; 8) (g + ^^ (60) 



CASE 3: COMMON UNKNOWN LOCATION PARAMETER ^ ^ 



In this case the more conservative procedure is to proceed under the 

 assumption that </ = 0. By the discussion above the probability require- 

 ment will in most problems be satisfied for all ^ ^ 0. The OC and ASN 

 functions, which are now functions of the true value of g, were already 

 obtained above. Of course, we need not consider values of g greater than 

 the smallest observed lifetime of all units tested to failure. 



Addendum 2 



For completeness it would be appropriate to state explicitly some of 

 the formulas used in computing the tables in the early part of the paper. 

 For the nonsequential, nonreplacement rule Ri with /c = 2 the proba- 

 bility of a correct selection is 



P(a; R,) = [ [ Mu, OAfrix, 6,) dy dx (61) 



where 



fXx, e) = '- C(l - e^'"y-' e-^^"-^+^"^ (r ^ n) (62) 



and C" is the usual combinatorial symbol. This can also be expressed in 

 the form 



P{a; R,) = 1 - (rC:r Z ^~^^"' 



;=i n - r -\-j (63) 



C'-l{B[r, n-r+l+a(n-r+ j)]}-' 



where B[x, y] is the complete Beta function. Eciuation (66) holds for 

 any g ^ 0. 



For the rule Ri the expected duration of the experiment for k = 2 

 is given by 



E{T) = r x{fr(x, d,)[l - Frix, 62)] + frix, d,)[l - Fr(x, ^i)] } dx (64) 



•'0 



where frix, 6) is the density in (62) and Fr{x, B) is its c.d.f. This can 



