MATHEMATICAL ANALYSIS OF RANDOM NOISE 321 



where p(K) is tlie probability of exactly K electrons arriving in the inter- 

 val (0, D, 



P{K) = ^-^e-' (1.1-3) 



and 



K 



I 



k=l 



hit) = t.ni - /a) (1.3-1) 



Multiplying /a-(/) and /k(/ + t) together and averaging li , h , • • ■ Ir 

 over their ranges gives 



/.(/)/.(/ + r) = Zi: r^--- f %i^(/-/AW/ + r-U 



A;=l m=l ^0 ^ Jo I 



This is similar to the expression for /|(/) which was used in section 1.3 to 

 prove Campbell's theorem and may Le treated in much the same way. 

 Thus, if t and / + t lie between A and T — A, the expression above becomes 



£" F(l)F{t + r)dl + ^'^^^3 ^^ £" F{1) dlj 



When this is placed in (2.6-1) and the summation performed we obtain 

 an expression independent of T. Consequently we may use (2.5-5) and get 



■ ^(r) = u f F{l)F(t + r) rf/ + 7(0' (2.6-2) 



J— 00 



where we have used the expression for the average current 



Z+OO 

 F{0 dt (1.3-4) 



GO 



In order to compute w{f) frcm ^(r) it is convenient to make use of the 

 fact that i/'(t) is always an even function of r and hence (2.5-1) may also 

 be written as 



/+» 

 Ut) cos IttJt dr (2.6-3) 



00 



Then 



dt F{t) / dr Fit + t) cos 27r/T 



00 J— 00 



Jlf)' COS iTrfrdr 



00 



