MATHEMATICAL ANALYSIS OF RANDOM NOISE 301 



All of the K characteristic functions are the same and hence, from (1.4-3), 

 PKiDdl is 



"^^ h ir '""" {f r ^'p ^'"^^^ ~ ^^^ ^o'' "^^ 



Although in deriving this relation we have taken A' > 0, it also holds for 

 K = (provided we use (1.4-4)). In this case Po{l) = 5(1), because 7 = 

 when no electrons arrive. 



Inserting our expression for P/r(/) and the expression (1.1-3) for p(K) 

 in (1.4-2) and performing the summation gives 



Pil) = ^ [ expl-ilii - uT 



+ V J exp {mF{t - t)] dAdu (1.4-5) 



The first exponential may be simplified somewhat. Using 



pT = u [ 

 Jo 



permits us to write 



— vT+v I exp [iuF{t — t)] dr = u i (exp [iuF{i — t)] — 1) dr 

 Jq Jo 



Suppose that A < / < T — A where A is the range discussed in connection 

 with equation (1.3-1). Taking | /^(Z - t) | = for | / - r | > A then 

 enables us to write the last expression as 



u f^'k"'^^'^ - l]dl (1.4-6) 



Placing this in (1.4-5) yields the required expression for P(I): 



P{I) = ^ / exp (-z7« + V J [e'"^'" - 1] dljdu (1.4-7) 



An idea of the conditions under which the normal law (1.4-1) is ap- 

 proached may be obtained from (1.4-7) by expanding (1.4-6) in powers of 

 11 and determining when the terms involving u and higher powers of ti 

 may be neglected. This is taken up for a slightly more general form of 

 current in section 1.6. 



dr 



jQ 



