calculating MGFs

**brogers** · February 7th 2011, 11:29 AM

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

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

b) Suppose that Xi is $Gamma(\alpha , \beta)$ . Find the MGF of X:
I tried MGF(t)= $(\frac{\beta}{\beta -t})^{\alpha\lambda}$ because $E(N)=\lambda$ 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!

**harish21** · February 7th 2011, 02:29 PM

can you show your work for part (a)?

(b)for independent random variables:

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

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

**brogers** · February 7th 2011, 04:44 PM

Originally Posted by harish21

can you show your work for part (a)?

(b)for independent random variables:

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

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

For E(X), I think it's easy to see.
$Var(X)=E(Var(X|N))+Var(E(X|N))=E(N^2 \sigma ^ 2)+Var(N \mu) = \lambda \sigma^2 + \mu^2 \lambda$

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

**matheagle** · February 7th 2011, 10:18 PM

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

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

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

**mr fantastic** · February 8th 2011, 12:55 AM

Originally Posted by brogers

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

[snip]

b) Suppose that Xi is $Gamma(\alpha , \beta)$ . Find the MGF of X:
I tried MGF(t)= $(\frac{\beta}{\beta -t})^{\alpha\lambda}$ because $E(N)=\lambda$ 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

**brogers** · February 10th 2011, 09:24 PM

Got it!! The pdf really helped. Thanks!

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