31. Give another proof of Exercise 27 by computing the moment generating function of Y^=\Xi a nd then differentiating to obtain its moments.

Hint: Let

ö(ß) = Å e x p ^ X * ; )

e x p ^

Now, exp / Ó Xi Í = ç = Å

Í

exp / Ó Xi = (Ö÷(0)ç

since Í is independent of the A^s where x(t) = E[etx] is the moment generating function for the A^s. Therefore,

ö(ß) = Å[(ö÷(0)Í]

Differentiation yields

ö\ß) = Å[Í(ö÷(0)Í-éÖ÷(01

"(/) = E[N(N - \)(Ö÷(ß))Í-\Ö'÷(ß))2 + Í(ö÷(0)Í-éÖ÷(0]

Evaluate at t = 0 to get the desired result.