1. ## calculating MGFs

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 $\displaystyle Var(X_{i})=\sigma^2$

a) Find E(X), var(X): I got this part with $\displaystyle E(X)= \lambda \mu$ and $\displaystyle var(X)= \lambda ^2 \sigma ^2 + \mu ^2 \lambda$

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!

2. can you show your work for part (a)?

(b)for independent random variables:

$\displaystyle E[e^{tX}]= E[e^{tX_1}\cdot e^{tX_2}\cdots e^{tX_n}]= E[e^{tX_1}]\cdot E[e^{tX_2}]\cdots E[e^{tX_n}]$

where $\displaystyle X_1, X_2,...X_n$ are random variables following a gamma distribution.

3. Originally Posted by harish21
can you show your work for part (a)?

(b)for independent random variables:

$\displaystyle E[e^{tX}]= E[e^{tX_1}\cdot e^{tX_2}\cdots e^{tX_n}]= E[e^{tX_1}]\cdot E[e^{tX_2}]\cdots E[e^{tX_n}]$

where $\displaystyle X_1, X_2,...X_n$ are random variables following a gamma distribution.
For E(X), I think it's easy to see.
$\displaystyle Var(X)=E(Var(X|N))+Var(E(X|N))=E(N^2 \sigma ^ 2)+Var(N \mu) = \lambda \sigma^2 + \mu^2 \lambda$

EDIT: now that I think about it, should it be $\displaystyle \lambda \sigma^2 + \mu^2 \lambda$?

4. I believe you need $\displaystyle E_N\bigl((1-\beta t)^{-\alpha N}\bigr)$

Try the substitution $\displaystyle e^s= (1-\beta t)^{-\alpha}$

So you have to figure out $\displaystyle E(e^{sN})=e^{-\lambda (e^s-1)}=e^{-\lambda[(1-\beta t)^{-\alpha}-1]}$

5. Originally Posted by brogers
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 $\displaystyle Var(X_{i})=\sigma^2$

[snip]

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!
pp17-18 here have the formula you need to use: http://www.markirwin.net/stat110/Lecture/Section45.pdf

6. Got it!! The pdf really helped. Thanks!