MATHEMATICAL ANALYSIS OF RANDOM NOISE 297 



Actually, as the equations (1.2-2) and (1.2-3) of Campbell's theorem 

 show, these averages and all the similar averages encountered later turn 

 cut to be independent of the time. When this is true and when the M in- 

 tervals in (1.2-5) are taken consecutively the time average (1.2-4) and the 

 average (1.2-5) become the same. To show this we multiply both sides of 

 (1.2-5) by di' and integrate from to T: 



I{t') = Lin-.it -i- i; ^I{1') dl' 



M-*ao M 1 »i=l ''0 



1 r^ ^^-^'^^ 



= Limit -— / 7(0 dl 



and this is the same as the time average (1.2-4) if the latter limit exists. 



1.3 Proof of Campbell's Theorem 



Consider the case in which exactly /I electrons arrive at the anode in an 

 interval of length T. Before the interval starts, we think of these K elec- 

 trons as fated to arrive in the interval (0, T) but any particular electron is 

 just as Ukely to arrive at one time as any other time. We shall number 

 these fated electrons frcm one to K for purposes of identification but it is to 

 be emphasized that the numbering has nothing to do with the order of ar- 

 rival. Thus, if tk be the time of arrival of electron number k, the probability 

 that tk lies in the interval (/, / -{- dl) is dl/T. 



We take T to be very large compared with the range of values of I for 

 which F(l) is appreciably different from zero. In physical applications 

 such a range usually exists and we shall call it A even though it is not very 

 definite. Then, when exactly K electrons arrive in the interval (0, T) the 

 effect is approximately 



K 



IkO) = lLFil- Ik) (1.3-1) 



the degree of approximation being very good over all of the interval except 

 within A of the end points. 



Suppose we examine a large number M of intervals of length T. The 

 number having exactly K arrivals will be, to a first approximation M p{K) 

 where p(K) is given by (1.1-3). For a fixed value of / and for each interval 

 having K arrivals, I Kit) will have a definite value. As M — > co , the average 

 value of the TK(tys, obtained by averaging over the intervals, is 



Jo I Jq I k=i 



= E/ '-pFit-lk) 

 k=i Jo I 



