Let $\{N_t:t \ge 0 \}$ be a Poisson process with rate $\lambda$. Find the followings:
(a) $\mathrm{Cov}(N_{3t}, N_{5t}|N_t)$
(b) $\mathbb E[\mathrm{Cov}(N_t, N_{3t}|N_{5t})]$
To find (a), using that $N_{3t} $ and $N_{5t}-N_{3t}$ are independent when given $N_t$, $\mathrm{Cov}(N_{3t}, N_{5t}|N_t)=\mathrm{Cov}(N_{3t}, N_{5t}-N_{3t}+N_{3t}|N_t)=\mathrm{Cov}(N_{3t}, N_{5t}-N_{3t}|N_t)+\mathrm{Cov}(N_{3t}, N_{3t}|N_t)=\mathrm{Var}(N_{3t}|N_t)=\mathrm{Var}(N_{3t}-N_t+N_t|N_t)=\mathrm{Var}(N_{3t}-N_t|N_t)+\mathrm{Var}(N_{t}|N_t)$
Where the last equality is from the fact that $N_{3t}-N_t$ and $N_t$ are independent.
Also, Since $N_t$ is already given, $\mathrm{Var}(N_{t}|N_t)=0$, and $\mathrm{Var}(N_{3t}-N_t|N_t)=\mathrm{Var}(N_{3t}-N_t)=2\lambda t$. Am I right?
For (b), since $N_{5t}$ is given, we cannot argue that $N_t$ and $N_{3t}-N_t$ are independent because they cannot exceed $N_{5t}$. What should I do for (b)?