SELECTING THE BEST ONE OF SEVERAL BIXOML^L POPULATIONS 567 



PcsiCB) 



(AlO) 



2(1)=0 



= P{X(2) < Xd)} + iP{X(2) = Za)} (All) 



= PcsiDB) (A12) 



The above is easily generalized to hold for any k > 2. The details of this 

 generalization are omitted. For general k this equality holds not only 

 for the important special case pn] > p[2] but also for the more general 

 case (2) for any t < k. Since the latter result is not needed here, the 

 proof is omitted. 



If we let G(y;p) denote the c.d.f. of the continuous binomial then 

 lemma 1 can be restated in the following form. 



Lemma 2: For any integer n and any y, the function G{y;p) is a non- 

 increasing function of p. In particular, for — | < y < n -\- ^ it is a 

 strictly decreasing fuyiction of p. 



Proof: For any y, set x = x(y) and d = d(y) equal to the integer part 

 and the fractional part of (y -\- h), respectively. Then for any y we have 

 the identity in p 



G(y;p) = H(x;p, d) (0 ^ p ^ 1) (A13) 



For any ^o such that — | < //o < w + | we have ^ x{yo) ^ n and 

 ^ diijo) S 1- The inverse function y{x,d) = x -\- 6 — | is a single- 

 valued function of the pair {x,d); the two particular pairs (0,0) and 

 (w,l) correspond to the unique values y = —^ and y = n -\- ^, re- 

 spectively. Hence the pair [x{yo), ^(^o)] must be different from these two 

 particular pairs above since it corresponds only to yo which is in the 

 interior of the interval ( — Ij ^ + h)- Lemma 2 then follows from lemma 1 

 and the fact that G(y;p) is identically zero in /; fei- y ^ —h and identi- 

 callj^ one in /) for ?/ ^ n + |. 



The probabilitj^ Pes of a correct selection for k discrete or k continuous 

 binomials for any configuration with p[i] > pi2] can now be A\Titten as 



/«+l/2 r k -j 



n P{Yu^ < 2/(1)} g(ya) ; Pm) dy^) (A14) 

 1/2 Lt=2 J 



/7!+l/2 r k 

 n G{y;p[;]) g(:y;p[i]) dy. (A15) 



1/2 L»=2 



Clearly if any one or more of the p[,] (i ^ 2) decreases, holding p^] 

 fixed, then it follows from lemma 2 that the right member of (A 15) is 



