Insurance claims from a portfolio of policies arise through time, one after another, according to a Poisson process with a constant rate of 28 claims per week. Claim sizes are also random variables, with each one assumed to be normally distributed, with a mean of $3200 and a standard deviation of $600, and with the claim size assumed to be independent of the times of occurrence and the accumulated numbers of claims.
What is the expected total size of all claims in any given week?
I tried to calculate the MGF but i get a strange result:
MGF= 
With a strange result. Then i tried to calculate the total size and i just multiplied 28 (claims) for 3200 ( the mean) = 89600
I'm sure i'm missing something. Is any of this two procedures correct? am I missing something?
Thanks
Best Answer
The second solution is correct but you have not provided any reason. It is either a consequence of Wald's formula (which the wiki article even has a subsection on exactly this class of problems in actuarial science), or one can prove it using properties of conditional expectations.
In your setting you have $N$ distributed like Poisson $(\lambda)$ where $\lambda=28$ and $(X_n)$ distributed Normal with mean $\mu=3200$ and $\sigma=600$ and the sequence $(X_n)$ is independent and $N$ is independent of all the $X_n$. The total size of the claims is the sum of all claim sizes for each claim that occurred: $$Y=\sum_{i=1}^N X_i.$$ The conditional expectation given $N=k$, is $$\mathbb{E}(Y|N=k)=k\mu,$$ by linearity of expectation. Thus, taking the expectation with respect to all possible $N=k$, we have $$\mathbb{E}(Y)=\mathbb{E}(\mathbb{E}(Y|N=k))=\lambda \mu=28\cdot 3200= 89600,$$ as you write. This is a very useful property of conditional expectations, by the way.