Policy holders of an insurance company have accidents at times of a Poisson process with rate $\lambda$. The distribution of the time $R$ until a claim is reported is random with $P(R ≤ r) = G(r)$ and $ER = ν$. (a) Find the distribution of the number of unreported claims. (b) Suppose each claim has mean $μ$ and variance $σ^2$. Find the mean and variance of the total size $S$ of the unreported claims.
Attempt. $(a)$ Thinning would be my first thought, but I would need an rv from the Bernoulli distribution, being $+1$ if the claim is reported (with probability $p=?$) and $0$ if the claim is unreported (with probability $1-p$). I am having difficulty finding $p$.
$(b)$ Here we use the definitions, since $\displaystyle E(S^m)=\sum_{k=0}^{\infty}k^mP(S=k)$ ($m=1,2$) and $V(S)=E(S^2)-E(S)^2.$
Thank you in advance.