pé) =| IY] p(é, 0, ¥Y) dY. (12.149) 
0 
In the same way as we did for the joint probability density p(Y, Y) in Section 12.7.1, 
relations between the joint moment generating function (it, it, it) and the joint cumulant 
generating function K(it, iz, it’) were used to give 
P(it, it, it’) = | | | p(y, Y,Y)exp[itY + it¥ + it¥] dy aY dY 
=1+>, ea Gt) Ge)”. (12.150) 
& jim! 
Here {jem is the joint moment 
[TS VONE = | | | yy'y"py,Y,¥) dY dY d¥ (12.151) 
mice 5 BD 60d Kiem put eat ise <2" 
K(it, if, i?) = log (it, it, i?) = > Fe Gi) GFY". 42.152) 
PRINS aes 
Therefore 
P(it, it, if) = exp [K(it, it, if )] (12.153) 
where j, k, and m are positive integers whose sum is greater than zero. Therefore, the 
inverse transform of Eq. 12.150, from Eqs. 12.150 and 12.152, is 
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