MATHEMATICAL ANALYSIS OF RANDOM NOISE 91 



and of the fact that the integrand is symmetrical in x and y. Integrating 

 dJ/dT with respect to T from to T\, using the formula 



r dT I fix) dx ^ r {Ty - x)f{x) dx, 

 Jo Jo Jo 



noting that / is zero when T is zero, and dropping the subscript on Ti finally 

 gives 



(E - £)' = 48 f dx [ dy(T - x)^p{x)^p{y)^P(x - y). 

 Jo Jq 



E* may be treated in a similar way. It is found that 



(E - EY - 3{E -£)'' = 3!2' [ dh [ dU [ dh [ dU^ly,,rPnh2h:i 



Jo Jo Jq Jo 



which may be reduced to the sum of two triple integrals. It is interesting 

 to note that the expression on the left is the fourth semi-invariant of the 

 random variable E and gives us a measure of the peakedness of the dis- 

 tribution (kurtosis). Likewise, the second and third moments about the 

 mean are the second and third semi-invariants of E. This suggests that 

 possibly the higher semi-invariants may also be expressed as similar multiple 

 integrals. 



So far, in this section, we have been speaking of the statistical constants 

 of E. The determination of an exact expression for the probability density 

 of E, in which T occurs as a parameter, seems to be quite difficult. 



When T is very small E is approximately / (t)T. The probability that 

 E lies in dE is the probability that the current lies in — /, — / —dl plus the 

 probability that the current lies in I, I -\- dl: 



2dl P E 



Vm. ^^P '^r (2-^«£r)-" exp -• — dE (3.9-14) 



where E is positive, 



r = {fj\ di==l{ETr"dE 



and T is assumed to be so small that /(/) does not change appreciably during 

 an interval of length T. 



Wlien T is very large we may divide it into a number of intervals, say n, 

 each of lengtli T/n. Let Er be the contribution of the r th interval. The 

 energy E for the entire interval is then 



£ = £i + £2 + • • • + £» 



If the sub-intervals are large enough the £r's are substantially independent 

 random variables. If in addition n is large enough E is distributed nor- 



