MATHEMATICAL ANALYSIS OF ILiXDOM NOISE 93 



and if we drop the subscript T and substitute the value of a in (3.9-18) we 

 get 



(f)" 



exp(-!^)^(^-^), n = i-l (3.9-20) 



r(« + 1) '^ \ a- / \a- /' a 



An idea of how this distribution behaves may be obtained from the 

 following table: 



24 25 21.47 24.67 28.17 .870 1.14 



48 50 44.1 48.7 53.5 .905 1.10 



where n is the exponent in (3.9-20). The column T(fb —fa) holds only for a 

 narrow band pass filter and was obtained by reading the curve yu in Fig. 1 

 of the above mentioned paper. The figures in this column are not very 

 accurate. The next three columns give the points which divide the dis- 

 tribution into four intervals of equal probability: 



^.25 = — ~ , -E.25 = energy exceeded 75% of time 

 ^.50 = — ^ , £.50 = energy exceeded 50% of time 



X 75 = - — -^ , £ 75 = energy exceeded 25% of time 



The values in these columns were obtained from Pearson's table of the in- 

 complete gamma function. The last two columns show how the distribu- 

 tion clusters around the average value as the normal law is approached. 



For the larger values of n we expected the normal law (3.9-15) to be 

 approached. Since, for this law the 25, 50, and 75 per cent points are at 

 f)i — .675cr, m, and m -f- .675cr we have to a first approximation 



x.,0 = % = (n + l) ^ T{f, - fa) 

 a- 



m ._ , A7C /— (3.9-21) 



^.25 = -, (w — .6/5cr) = x.oo — .o75v^£o 



^.75 = X,50 -f .675\/x.5o 

 This agrees with the table. 



