566 THE BELL SYSTEM TECHNICAL JOURNAL, \L\.RCH 1957 



k 



Pes = P{X(o < Xa) for t ^ 2} + i J^ P{Z(„) = Zo) and 



X(o < Xd) for i ^ 2, i ^ a} + " • (A5) 



+ T-P{X(1) = X(2) = ••• = X{k)} 



It will be necessary to write the Pes for any configuration with p[i] > p[2] 

 in another form which is more useful for the purpose at hand. Corre- 

 sponding to an}' binomial chance variable A' (which takes on integer 

 values from to n) we define a "Continuous Binomial" chance variable 

 Y by letting Y be uniformly distributed in the interval (j — 2) i + 2) 

 with the same total probabiHty in this interval as the ordinary binomial 

 assigns to the integer j, namelj'' 



C/V(i -py~' (i = o, 1, ..-,71) 



We will now show that the probability Pc^ of a correct selection is unal- 

 tered if we replace each of the k discrete binomials b}' its corresponding 

 continuous binomial. Let F(o denote the continuous binomial (CB) 

 chance variable associated with p[i] and let ya) denote any value it can 

 take on. Let X(,) denote the nearest integer to F(o and let i\i) denote 

 the nearest integer to y^i) {i = 1, 2, • • • , A;). Then X(,-) is a discrete bi- 

 nomial (DB) with the same parameters (/)[,] , n). Let 



g{x, p) = CxV(l - pr-' (X- = 0, 1, . • ., n) 



Then the density g(y, p) of the continuous binomial (disregarding the 

 half-integers) is given by g{y, p) = g(x, p) where x is the nearest integer 

 to y. 



For two continuous binomials (i.e., k = 2) the probability Pes of a 

 correct selection for any configuration with p[i] > p[2] is given bj' 



/n+l/2 

 P{Yi2) < 2/(1)1^(^(1) ;P[i])dya) (A6) 



1/2 



" /-5;(l) + l/2 



= J2 / P{y{2) < y(i)]g(ya) -yPii]) dya) (A7) 



a;(l)=0 Jx(i)-l/2 



(A8) 



Within any interval (x^) — h, •'C(i) + h) ^'6 have 

 P{Y,2) < yo)} = P{X(2) < Xa)} 



-f P{Z(2) = a-a)}P{F(2) < ya) 1 X(2) = .T(i)} 

 = P{X(2) < Xa)} + h P{X<2) = Xa)} (A9) 



which depends only on Xa) . Hence from (A7) 



