MATHEMATICAL ANALYSIS OF RANDOM NOISE 303 



where q(a) is the probabiHty density function for the a's. It turns out that 

 the probability density P(I) of / as defined by (1.5-1) is 



1 r^" / r^°° 



P{I) = TT- I exp I —iln + V I q{a) da 

 2ir J— 00 \ J— 00 



The logarithm of the characteristic function of F(I) is, from (1.5-4), 

 V [ q(a) da I " [e'"''''^'^ - \\dt 



J— 00 J— 00 



n=l Wl J— 00 J— 00 



Comparison with the series (1.5-3) defining the semi-invariants gives the 

 extension of Campbell's theorem stated by (1.5-2). 



Other extensions of Campbell's theorem may be made. For example, 

 suppose in the expression (1.5-1) for 1(1) that ti , h , - • ■ tk , • • - while still 

 random variables, are no longer necessarily distributed according to the 

 laws assumed above. Suppose now that the probability density p{x) is 

 given where x is the interval between two successive events: 



t2 = h-\- xi (1.5-5) 



^3 = ^2 + ^2 = ^1 + :Vi + X2 



and so on. For the case treated above 



p(x) = ve-'\ (1.5-6) 



We assume that the expected number of events per second is still v. 

 Also we take the special, but important, case for which 



F{t) = 0, / < (1.5-7) 



F{t) = g-"', / > 0. 



For a very long interval extending from t = iitot = T •{- ti inside of which 

 there are exactly K events we have, if / is not near the ends of the interval, 



/(/) = aiF(t - h) + a2F{t - h - x^) + - - • 



+ aK+iF{t — h — xi'" — Xk) 



= aiF(/') + 02F(/' - .Ti) H + OK^iFit' - xi- ■■■ - Xk) 



