I'm trying to answer the following question:
"A person collects coupons one at a time, at jump times of a Poisson process $(N_t)_{t\geq 0}$ of rate $\lambda$. There are m types of coupons, and each time a coupon of type j is obtained with probability $p_j$, independently of the previously collected coupons and independently of the Poisson process. Let T be the time the complete set is obtained.
Show that $\mathbb{P}(T I'm able to find the distribution of T, by splitting the process into independent subprocesses of rate $\lambda p_i$, but am not sure where to go for the second part. It seems that explicit computation $\mathbb{E}L$ or $\mathbb{E}[L|T]$ is infeasible. Is it possible to use the fact that $T$ is a stopping time here and perhaps the strong Markov property? Thank you.