The variable S has a compound Poisson claims distribution with the following:
- Individual claim amounts equal to $1$, $2$, or $3$.
- E(S) = $56$.
- Var(S) = $126$
- $\lambda = 29$
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?