84 BELL SYSTEM TECHNICAL JOURNAL 



APPENDIX I 



Cumulative Distribution Function for a Sum of Squares of Normal 



Variates 



Let .V be a random variable defined by 



M 



x= Y. Jn (AM) 



71 = 1 



where y,, is a random variable distributed normally about its average value 

 jn with unit standard deviation. In writing {A\ — \) we have been guided 

 by (4-3), where M = 2N + 1, but here we shall let M be any positive integer. 

 In much of the following work M/2 occurs and for convenience we put 



m = M/2 (Al-2) 



From the work of Section 4 it follows that the probability density p{x, «) 

 of X is given by Fisher's expression 



p{:x, u) = 2-'{x/u)^i-'-^i-' /^_:[(zix)i/2]e-("+-)/2 (^1-3) 



where u is the constant 



n 



E fn (Al-4) 



71 = 1 



Here we are interested in the cumulative distribution function, i.e., the 

 probability that x is less than some given value xq, 



P(xo, «) = [ p(x, n) dx (A 1-5) 



as M becomes large. In this case the central limit theorem tells us that 

 p{x, u) approaches a normal law with average x = M -\- n and variance = 

 ave. (x — x)- = 2M + 4u. The function P(.Vo, u) has been studied by J. I. 

 Marcum in some unpublished work, and by P. K. Bose(9). In i)articular, 

 Marcum has used the (iram-Charlier series to obtain values for P(.Vo, u) in 

 the vicinity of x for large values of M. However, since I have not been able 

 to find any previous work covering the case of interest here, namely values 

 of P(xo, u) when ;Vo is appreciably less than x, a separate investigation is 

 necessary and will be given here. 



Integrating the general expression (4-5) with rcsj)cct to .v between — A^ 

 and .Vr,, letting X— > co, and discarding the portions of the integrand which 

 oscillate with infinite rapidity gives 



