90 BELL SYSTEM TECHNICAL JOURNAL 



where, from (3.9-3) 



Mr = WoTifb - fa) (3.9-11) 



The third moment E^ may be computed in the same way. However, in 

 this case it pays to introduce the characteristic function for the distribution 

 of I(ti), lih), I{k). Since this distribution is normal its characteristic 

 function is 



Average exp [izili + izili + izzh] 



= exp - y (zi + 22 + zl) + ypih - h)ziZ2 .^ ^_^^. 



] 



■i- 4'ih — tijZiZs + Xpits — /2)Z2 23 



From the definition of the characteristic function it follows that 



2 2 2 



Illlll= -coeff. Of ^-i|^; in ch. f. 



= l/'O + 2\po{\p2i + V'31 + ^32) 

 + S\p2l4^Sl4^32 



where we have written i/'2i for i/'(/2 — ^1), etc. When (3.9-13) is multiplied 

 by dti dti dtz , the variables integrated from to T, and the above double 

 integral expression for ar used, we find 



(E - Ef = 2\t [ dh [ dU [dh^l^.2lhl^■s2. 

 Jq Jq Jo 



Denoting the triple integral on the right by / and differentiating, 



^ = 3 [ dh f dhHi2 - h)i{T - h)i{T - h) 

 dl Jo •'0 



= 3 / dx \ dy\p{x — y)4'{x)\p{y) 

 Jo Jo 



= 6 / dx i dy\l/(x — y)rl/(x)4'(y) 

 Jo Jo 



In going from the first line to the second ti and 1-2 were replaced by T — x and 

 T — y, respectively. In going from the second to the third use was made of 

 the relations symbolized by 



' dx \ dy = I dx I dy -{- I dx I dy 

 ^0 Jo Jo Jq *'x 



= / dx I dy -\- I dy I dx 

 Jo Jo Jo Jo 



