MATHEMATICAL ANALYSIS OF RANDOM NOISE 299 



Counting up the number of terms in the double sum shows that there are K 

 of them having the first value and A' —A' having the second value. Hence, 

 if A < / < r - A we have 



Averaging over all the intervals instead of only those having K arrivals 

 gives 



nt) = z p{K) i\{i) 



K=0 



+ 00 



=./ 



F\t) dt + /(/)2 



where the sunmiation with respect to K is performed as in (1.3-4), and after 

 summation the value (1.3^) for /(/) is used. Comparison with (1.3-5) 

 estabhshes the second part of Campbell's theorem. 



1.4 The Distribution of /(/) 



When certain conditions are satisfied the proportion of time which /(/) 

 spends in the range I, I -\- dl is P{I)dI where, as v -^ co , the probabihty 

 density P{I) approaches 



^ e-''-'^'"'] (1.4-1) 



<Ti\/2Tr 



where /is the average of /(/) given by (1.2-2) and the square of the standard 

 deviation o-/ , i.e. the variance of /(/), is given by (1.2-3). This normal 

 distribution is the one which would be expected by virtue of the "central 

 limit theorem" in probability. This states that, under suitable conditions, 

 the distribution of the sum of a large number of random variables tends 

 toward a normal distribution whose variance is the sum of the variances 

 of the individual variables. Similarly the average of the normal distribu- 

 tion is the sum of the averages of the individual variables. 



So far, we have been speaking of the hmiting form of the probability 

 density P(/). It is possible to write down an exphcit expression for P(/), 

 which, however, is quite involved. From this expression the limiting form 

 may be obtained. We now obtain this expression. In line with the dis- 

 cussion given of Campbell's theorem, we seek the probability density P{I) 

 of the values of l{i) observed at t seconds from the beginning of each of a 

 large number, M, of intervals, each of length T. 



