An insurance company receives i.i.d. claims $\displaystyle X_{1}, X_{2},...X_{N}$ where $\displaystyle N ~ Poisson(\lambda)$. Let $\displaystyle X=X_{1}+X_{2}+...X_{N}$. Suppose $\displaystyle E(X_{i})=\mu$ and $\displaystyle Var(X_{i})=\sigma^2$

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b) Suppose that Xi is $\displaystyle Gamma(\alpha , \beta)$. Find the MGF of X:

I tried MGF(t)=$\displaystyle (\frac{\beta}{\beta -t})^{\alpha\lambda}$ because $\displaystyle E(N)=\lambda$ and the sum of independent variables' MGF is just the product of each MGF, but the results doesn't match part (a). Please help me out with this part. Thanks!