# compound poisson

The variable S has a compound Poisson claims distribution with the following:

1. Individual claim amounts equal to $1$, $2$, or $3$.
2. E(S) = $56$.
3. Var(S) = $126$
4. $\lambda = 29$

Determine the expected number of claims of size 2.

I let X be the individual claim i.e. S = $X_1 + X_2 + ... +X_n$ where $\lambda = \lambda_1 + \lambda_2 + ... + \lambda_n$. But how do I find the first moment and second moment of X from S? Can I treat X as independent and identically distributed random variables to solve it?

$x_1,...,x_n$ are not i.i.d. but they are similar, they have the same pdf with different parameter and are independent. We know the moment generating function is given by $E[e^{(xt)}]$. Substituting in $S=\sum x_i$ for $x$ we get $E[e^{(St)}]=E[e^{(\sum x_i)t}]=(E[e^{(x_1t)}])(E[e^{(x_2t)}])...(E[e^{(x_nt)}])$. We know the moment generating function for each $x_i$ since we know its pdf and its given parameter $λ_i$. From there it's pretty straight forward I think.