Lmn 
i (10 :)"(i02)" exp|— i( 12) + 0222)|d0 1dO>. 
1 
P(2, 22) = (270) ee 
ee 
(12.107) 
This Eq. 12.107 shows that p(z;, 22) can be calculated if we know all the m,nth order 
moments of this probability function. 
On the other hand, by the definition of the cumulant 4,,, and the cumulant 
generating function K(i6;, i62). 
K(i61, 102) = log (61, 82) = at (16 :)(G62) + 2 (16;)(162) + 
Konan SRG PE TAMEATT 
= Sey 81) "(i82)". (12.108) 
Therefoie. 
2 2 kinn ° m5, nN 
(61,2) = exp{K (1, 102) = no So (162) 
mn min: 
kon 1 ; 
=1 {54s sa - - 8 rosy} Ai i {de ar warvo.r| 
tof oe oo ee 22109) 
When we insert this @(@;, 02) into Eq. 12.106, 
1 
P(21, 22) SCE | J elton. io) —1012. + 0r20}ae 
rai (12.110) 
J | exp <2 (i63)" (62) - oO:Z1+9222) 99 dO. 
Se & min! 
From Egs. 12.104 and 12.109, kp, and Um, are related as follows: 
356 
