I'm a new learner of conditional expectation. I got stuck on this problem: Suppose $\{N_t\}$ is a Poisson process with rate $\lambda$, then what is $E[N_s|N_t]$ for $s < t$?
My progress so far: Define $X_1 \sim Po(\lambda s)$ and $X_2 \sim Po(\lambda (t-s))$. $X_1$ and $X_2$ are independent. Then $E[N_s|N_t]=E[X_1|X_1+X_2].$
I know $E[X_1|X_1+X_2]=\phi(X_1+X_2)$ for some Borel function $\phi(x)$. Question is how to find it here? Another problem that has confused me quite long time is that what is $\sigma(X_1+X_2)$ for two independent RVs $X_1$ and $X_2$ here?