324 BELL SYSTEM TECHNICAL JOURNAL 



The product of N such probabilities is, as /V — > oo , A/ — ^ 0; 



exp -K \ p{l) dl = e~^ 



This is the probability that exactly events happen in T. In the same way 

 we are led to the expression 



^e-^ (2.6-9) 



A! 



for the probability that exactly K events happen in T. 



When we consider many intervals (0, T) we obtain many values of A' and 

 also many values of / measured / seconds from the beginning of each interval. 

 These values of 7 define the distribution of / at time t. By proceeding as in 

 section 1.4 we find that the probabihty density of / is 



P(7, t) = ^ [ du ex^\ -iul + K [ p{x)ie''^^'~'^ - I) dx\ 

 2ir J-x, l_ •'0 J 



The corresponding average and variance is 



I = K I p{x)F{t - x) dx 

 Jo 



{1 - If = K [ p{x)FHt - x) dx (2 6-10) 



Jo 



If S(f) is given by (2.1-2) and 5(/) by (2.6-5) (assuming the duration of 

 F{t) short in comparison with T) the average value of \ S{f) \ may be ob- 

 tained by putting (1.3-1) in (2.1-2) to get 



1 



Expressing SK(f) SAf), where the star denotes conjugate complex, as a 

 double sum and averaging over the /a;'s, using p{t), and then averaging over 

 the A's gives 



I Sif) f = K I 5(/) 1^ fl + A I jf "^ pix)e-''''' dx I'] (2.6-11) 



This may be used to compute the power spectrum from (2.5-3) provided 

 p(x) is not periodic. If p{x) is periodic then the method of section 2.2 

 should be used at the harmonic frequencies. If the fluctuations of p{t) are 

 slow in comparison with the fluctuations of F(i) the second term within the 

 brackets of (2.6-11) may generally be neglected since there are no values of 



