# Thread: mgf of a sum of a poisson number of iid rvs

1. ## mgf of a sum of a poisson number of iid rvs

Suppose that $X_1,X_2,...$ are i.i.d. random variables, each of which has m.g.f. $\psi(t)$. Let $Y=X_1 + ... + X_N$, where the number of terms $N$ in the sum is a random variable having the Poisson distribution with mean $\lambda$. Assume that $N$ and $X_1,X_2,...$ are independent, and $Y=0$ if $N=0$. Determine the m.g.f of $Y$.

I know the m.g.f. of $Y|(N=n\ne0)$ is $[\psi(t)]^n$.
I don't know where to go from there.

2. ## Re: mgf of a sum of a poisson number of iid rvs

Originally Posted by JJMC89
Suppose that $X_1,X_2,...$ are i.i.d. random variables, each of which has m.g.f. $\psi(t)$. Let $Y=X_1 + ... + X_N$, where the number of terms $N$ in the sum is a random variable having the Poisson distribution with mean $\lambda$. Assume that $N$ and $X_1,X_2,...$ are independent, and $Y=0$ if $N=0$. Determine the m.g.f of $Y$.

I know the m.g.f. of $Y|(N=n\ne0)$ is $[\psi(t)]^n$.
I don't know where to go from there.
$\psi_Y(t)=E(e^{tY})=\sum_{n=0}^{\infty} p(n)E(e^{tY}|N=n)=\sum_{n=0}^{\infty} p(n)[\psi(t)]^n$

(the mgf when $n=0$ is still $[\psi(t)]^n=1$)

CB