TITLE: Covariance and variance of a Poisson r.v.
QUESTION [5 upvotes]: Given a Poisson process $N(t),t\geq 0$ with rate $\lambda$ and another r.v. $T$ independent of $N(t)$ with mean $\mu$ and variance $\sigma^2$, I would like to compute the following quantities:
$$ \mathbb{Cov}(T,N(T)) \ \ \mbox{ and } \ \ \mathbb{Var}(N(T))$$
My guess is respectively: $\lambda \mu + \lambda \sigma^2$ and $\sigma^2\lambda$. But I am not sure it is correct nor how to justify some steps.
Anyone knows? Thank you very much!

REPLY [5 votes]: Since
$$
  \mathbb{Cov}\left(T, N(T)\right) = \mathbb{E}\left(T N(T)\right) - \mathbb{E}(T) \mathbb{E}\left(N(T)\right) = \mathbb{E}\left(T \, \mathbb{E}\left(N(T)\mid T\right)\right) - \mathbb{E}(T) \, \mathbb{E}\left(\mathbb{E}\left(N(T)\mid T\right)\right)
$$
But $\mathbb{E}\left(N(T) \mid T\right) = \lambda T$, thus
$$
  \mathbb{Cov}\left(T, N(T)\right) = \mathbb{E}\left(\lambda T^2\right) - \lambda \mathrm{E}(T)^2 = \lambda \mathbb{Var}(T)
$$
Similarly:
$$ \begin{eqnarray}
   \mathbb{Var}\left(N(T)\right) &=& \mathbb{E}\left( \mathbb{E}\left(N(T)^2|T\right) \right) - \mathbb{E}\left(\mathbb{E}\left( N(T) \mid T\right)\right)^2 \\ &=&
    \mathbb{E}\left( \lambda^2 T^2 + \lambda T\right) - \mathbb{E}\left(\lambda T\right)^2 \\ &=& \lambda^2 \mathbb{Var}(T) + \lambda \mathbb{E}(T)
\end{eqnarray}$$