Let $\{N_T:t\geq 0\}$ a homogeneous Poisson process with rate $\lambda\geq0$ and $T\geq0$ independent random variable with mean $\mu$ and variance $\sigma^2$. Find $cov(T,N_T)$.
How may I compute $cov(T,N_T)$? May I set T=t and use conditional expectation? I don't know how to proceed.
$$ COV[T,N_T] = E[T * N_T] - E[T] E[N_T] $$
We first find some expected values to solve for this
$$ E[N_T] = E \left[ E[N_T|T] \right] = E[\lambda T] = \lambda E[T] $$
$$ E[T N_T] = E[E[T N_T|T]] = E[T \lambda T] = \lambda E[T^2] $$
Hence,
$$ COV[T,N_T] = \lambda E[T^2] - E[T] \lambda E[T] = \lambda ( E[T^2] - E[T]^2) = \lambda \sigma^2 $$