566 BELL SYSTEM TECHNICAL JOURNAL 



The expansion given above for I{p) is of great practical value in 

 connection with the evaluation of the incomplete binomial sum- 

 mation. We know ^ that 



■ P{cvn,p) = E (")M1 - py-' 



x=c ^" ^ 



therefore, we have 



P(c; n, p) = , ^ ""l ^, Z f^— V^(m, c - \)P{c + m, T^. 



This approximation formula is submitted for ranges of values of c 

 and n such that c is of the order of magnitude of V'^ or less. But for 

 values of P{c\ n, p) which are of most interest, say, 



.0001 < P < .9999, 



c is of the same order of magnitude as 



a = np 



when 11 is a large number. Therefore, the formula is particularly 

 valuable for values of p which are of the same order of magnitude as 

 1/Vw or less. 



The first table at the end of this paper indicates the degree of 

 accuracy obtainable by taking d = and using only one or two 

 terms of the approximation for P{c; n, p). The table also indicates 

 the result of taking d such that Ti = a = np, and using only one, 

 two, three or four terms of the approximating series. 



I am indebted to Miss Elizabeth McCusker and Miss Catherine 

 Lennon of the Department of Development and Research of the 

 American Telephone and Telegraph Company for the computation 

 work involved in the construction of the tables at the end of this 

 paper. 



II 



Consider the Bessel Function 



^"(*) = /-w!^i 1^ r (^^"^)'" ^'^""^ ^•^'' 



V ivT(n + 2) Jo 

 ^ See Laplace, " Theorie Analytique des Probabilites," 1st edition, p. 151. 



