SELECTING THE BEST ONE OF SEVERAL BINOMIAL POPULATIONS 565 



Appendix II 



MONOTONICITY PROPERTIES 



We shall prove that for any fixed d (0 ^ d ^ 1) the probability Pes 

 of a correct selection is smaller for the configuration: 



P[l] - d = P[2] = P[3] = ■■■ = pik] (Al) 



than for any configuration given by 



P[i] - d ^ p[2] ^ p[s] ^ • • • ^ Pik] (A2) 



where pn] is considered fixed and the p^] (i ^ 2) are variables. In other 

 words, for fixed p[i] the probability Pes is a strictly increasing function 

 of each of the differences 



Pm - P[i] ii ^ 2) 



We shall need the following lemma. 



Lemma 1: For any pair of integers x, n (0 ^ x ^ n) and any 

 6 {0 S d ^ 1), not depending on p, the function 



Hix; p, d) = i; crpxi - pY-' + ec:p'{i - py-^ (as) 



is a decreasing function of p over the unit interval (0 ^ p ^ 1). More- 

 over, it is strictly decreasing unless (x = and 6 = 0) or (x = ?i a7id 



e = i). 



Proof: Differentiating (A3) with respect to p gives after telescoping 

 terms 



{6 - l)xCxV(l - p)""" - d(n - .r)C.V(l - p)"~"~' (A4) 



which is negative for < p < 1 unless (x = and 6 = 0) or (x = n and 

 6 = 1). Since ^ is continuous in p at p = and p = 1 the lemma follows. 

 Let X(i) denote the chance number of successes that arises from the 

 binomial process associated with 



Pii] (*■ = 1, 2, • • • , n) 



the value of the integer n is assumed to be fixed throughout this discus- 

 sion and it will usually not be listed as an argument. The probability 

 Pes of a correct selection for any configuration with p^j > p[2] is 

 given by the expression on the top of the next page. 



