284 BELL SYSTEM TECHNICAL JOURNAL 



where v is the average number of electrons arriving per second at the anode. 

 In this expression the electrons are supposed to arrive independently and at 

 random. ve~*'' dt is the probability that the length of the interval between 

 two successive arrivals lies between / and / + dt. 



2. Generalization of Campbell's theorem. Campbell's theorem gives 

 information about the average value and the standard deviation of the 

 probabihty distribution of /(/). A generalization of the theorem gives 

 information about the third and higher order moments. Let 



m = T.akF(t - 4) (1.5-1) 



— 00 



where F(t) and tk are of the same nature as these in (1.2-1) and • • -ci , 

 az , ■ • ' dk , • • • are independent random variables all having the same 

 distribution. Then the n' semi-invariant of the probability density P{I) 

 oil = I{t) is 



X„ = ,? f '^ [F{t)Tdt (1.5-2) 



J— 00 



The semi-invariants are defined as the coefficients in the expansion of the 

 characteristic function /(«/): 



\ogef{u) = S -: ("0" (1.5-3) 



where 



f{u) = ave. /'" = f " P(/)e''" dl 



J—eo 



The moments may be computed from the X's. 



3. As V —^ <x> the probabihty density P(I) of the shot effect current ap- 

 proaches a normal law. The way it is approached is given by 



-1 (0)/- \ A3<r (3). 



P(/)-a-^V^^"^(x)-l^<,-(^) 



+ 



—^ <P ix) + -^ if {x) \-\- 



(1.6-3) 



where the X's are given by (1.5-2) and 



Since the X's are of the order of v, a is of the order of v^'^ and the orders of 

 (r~', X3<7~*, X4(r~' and Xjo-"' are j'""''', v~\ v'^^"" and v~^'' respectively. A 



