200 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



SO that QilQi = 10. The expression (52) reduces to that in (44) if we set 

 di = 02 = 6 and replace a and h by n/2 in the resulting expression. 



Corresponding exact expressions for E(Ta.b,j) for j > 1 are extremely 

 tedious to derive and unwieldy although the integrations involved are 

 elementary. If we let g —^ oo then we obtain expressions for the non- 

 replacement case which are relatively simple. They are best expressed 

 as a recursion formula. 



E(.Ta,bj) = — , ,. ETa-\,b,}-l 



+ m^ ^"— ^^ = '^ 



(53) 



EiT.,b.d = "^^ ^ 



aXi + 6X2 (a — l)Xi + 6X2 



I 0X2 1 ( h > ^^ 



"^ aXi + 6X2 aXi + (6 - 1)X2 ' = 



(54) 



E(Tafij) ^ g + di/a fori ^ a and j = (55) 



E{Ta,oJ = dr/(a -j) for 1 ^ i ^ a - 1 (56) 



Results similar to (55) and (56) hold for the case a = 0. The above 

 results for gr = 00 provide useful approximations for E{Ta,b,j) when g 

 is large. Upper bounds are given by M 



E{Ta,bj) ^ [aXi + (6 - i)X2r (i = 1, 2, • • • , h) (57) 



E(Ta.bj+b) ^ [(a - j)Xr' (i = 1, 2, • . • , a - 1). (58) 



Duration of the Experiment 



For the sequential rule R^' with k = 2 we can now write down approxi- 

 mations as well as upper and lower bounds to the expected duration 

 E{T) of the experiment. From (50) 



I 



g + ..5^;^.\ s E(T) = E /?(r.,,) 



c-l 



n(Xi -f X2) ^ '''^ ' ~ § '^^^ "'"'^^ (59) 



+ \FA¥; 5) - c]i!;(T„,„,.) 



where c is the largest integer less than or equal to E{F\ 5). The right ex- 

 pression of (59) can be approximated by (53) and (54) if g is large. If 

 c < 2n then the upper bounds are given by (57) and (58). A simpler 



j 



