# Thread: MGF of a random sum

1. ## MGF of a random sum

Let $\displaystyle X_1, X_2,...$~N(0,1) be an iid sequence and let N ~ Poi$\displaystyle (\lambda)$ independently.

Find the MGF of the random sum

$\displaystyle S = \sum_{k=1}^N X_k$

What I do know is the pmf and pdf of the N and $\displaystyle X_n$ , but im confused with the random sum and how to calculate it. Can anyone please help?

I know that the MGF of the random sum is the product of the MGFs' of the RV's that make up that random sum.

2. Originally Posted by Maccaman
Let $\displaystyle X_1, X_2,...$~N(0,1) be an iid sequence and let N ~ Poi$\displaystyle (\lambda)$ independently.

Find the MGF of the random sum

$\displaystyle S = \sum_{k=1}^N X_k$

What I do know is the pmf and pdf of the N and $\displaystyle X_n$ , but im confused with the random sum and how to calculate it. Can anyone please help?

I know that the MGF of the random sum is the product of the MGFs' of the RV's that make up that random sum.
1. Find the moment generating function of the sum given N = n (you're accustomed to doing this).

2. Treat the above result as a function of n and take its expectation with respect to the distribution of N.

Read this: Compound Poisson distribution - Wikipedia, the free encyclopedia

Or this (p5): http://www.stats.uwo.ca/faculty/kulp...ndouts/MGF.pdf