Situation:
We have that $\{W_t,t \geq 0\}$ is a Brownian motion and $\{N_t,t\geq 0\}$ is a Poisson process such that $N_t$ follows a Poisson distribution with parameter $t$. This process is independent from our Brownian motion. We define the following stopping time: $\tau_n =\inf\{t\geq 0:N_t \geq n\}$.
Question: We are given the hint that we must use the fact that $N_t - t$ and $(N_t - t)^{2}-t$ are martingales, and that we must use Doob's optional sampling theorem. The exercise is to find:
$\mathbb{E}[(\tau_{n} - n)^2]$ and $\mathbb{E}[\tau_{n}]$
My attempt: I'm getting confused with the way the stopping time is defined, more specifically the subscript $n$ in there. I figured that from the optional sampling theorem follows that:
$\mathbb{E}[N_{t \wedge \tau_{n}}]=\mathbb{E}[t\wedge\tau_{n}]$
And then using the dominated convergence theorem and the monotone convergence theorem to find $\mathbb{E}[\tau_n]$, but this doesn't give correct results. This was pretty much my only idea on solving this exercise so I was going to use the same approach for finding the other expectation except then using the other martingale.
Tips on solving this exercise are much appreciated (rather than a full answer). Thanks in advance!