74 



BELL SYSTEM TECHNICAL JOURNAL 



From (3.6-8) and (3.6-5) we may obtain the probability density function 

 for the maxima in the case of a low pass filter. Thus the probability that 

 a maximum selected at random from the universe of maxima will he in 

 I, I -\- dl is 



dl 



3v 2x^0 _ 

 where 



2,-9.^/8 ^(StT 



1/2 



ye 



I 



1 +erfj(- 



l/2\ - 



(3.6-9) 



Fig. 2^Distribution of maxima of noise current. Noise through ideal low-pass filter. 

 -7= dl = probability that a maximum of / selected at random lies between I and I + H- 



When y is large and positive (3.6-9) is given asymptotically by 



dl \/5 -J,2;2 



— =. -^^ — ye 

 V'/'o 3 



If we write (3.6-9) as pi{y) dy, the probability density pi{y) of y may be 

 plotted as a function of y. This plot is shown in Fig. 2. The distribution 

 function P{I„u,^ < ys/xj/o) defined by 



P(/,nax < yV^o) = J Pi(y)-dy 



and which gives the probabihty that a maximum selected at random is 

 less than a specified y\/\l/o = I, is one of the four curves plotted in Fig. 4. 

 If / is large and positive we may obtain an approximation from (3.6-5). 

 We observe that 



\M 



(4) 



'Ao'/' 





