Poisson process -- conditional expectation problem

Question

Let ${N_t}$ be homogeneous Poisson process with rate $\lambda$.
The random variables $$X_i = N_{t_i} - N_{t_{i-1}}$$ are independent with marginal distributions $X_i$ ~ Poisson($\lambda(t_i - t_{i-1}))$.

Find: $$E(N_s|N_t)$$ and $$E(N_t|N_s)$$ for $s < t$.

Answer 1

Since $s\leq t$ we have that $N_s$ is $N_t$-measurable, thus $\Bbb E [N_s \vert N_t] = N_s$. Further $N_t - N_s$ is independent from $N_s$. Therefore, $$\Bbb E [N_t \vert N_s] = \Bbb E [ N_t - N_s \vert N_s] + \Bbb E[N_s \vert N_s]= \Bbb E [N_t - N_s] + N_s\\ = \lambda(t-s) + N_s ,$$ since $N_t - N_s \sim$ Poi$(\lambda (t-s)).$