Poisson Process with a Random Variable I really couldn't wrap my head around this basic concept so I'm looking help for some basic calculations to solidfy my understanding: 
Suppose we have $T\in (0,\infty)$. Say we have $E(T)=\mu$, $Var(T)=\sigma^2$, and it has MGF $M(t)$. $T$ is independent of the Poisson process $\{N(t),t>0\}$ with rate parameter $\lambda$. Consider the random variable $N(T)$. Then how do I find the following terms in terms of $\mu$, $\sigma^2$, and $M(t)$?


*

*$E[N(T)]$

*$Var[N(T)]$

*$Cov[N(T),T]$

*$P(N(T)>0)$


Thanks for any help! 

Let $X$, $Y$ independent random variables, then
$$\mathbb{E}(f(X,Y) \mid Y) = \mathbb{E}(f(X,y)) \bigg|_{y=Y}$$
and so by tower property
$$\mathbb{E}(f(X,Y)) = \mathbb{E} \big( \mathbb{E}(f(X,y)) \big|_{y=Y} \big)$$
In our case, $X \hat{=} N$, $Y \hat{=} T$ - they are independent by assumption. So, for example,
$$\mathbb{E}(N(T)) = \mathbb{E} \big( \mathbb{E}(N(t)) \big|_{t=T} \big) = \mathbb{E} \big( \lambda \cdot t \big|_{t=T} \big) = \lambda \cdot \mathbb{E}(T)$$
where we used that $N(t) \sim$ Poi($\lambda \cdot t)$ for the second equality. The other calculations are similarly.
