($N_t$) is Poisson process with $\lambda = 1$. Calculate $E(N_2\mid N_1)$ and $E(N_1\mid N_2)$

Question

($N_t$) is a Poisson Process with constant rate $\lambda = 1$.
$1)$ Calculate $E(N_2\mid N_1)$:

So this is how far I've gotten:

Let $N_2 = N_1 + (N_2 - N_1)$

$E(N_2\mid N_1) = E(N_1\mid N_1) + E(N_2 - N_1\mid N_1) = N_1 + E(N_2 - N_1)$

But then I get stuck and I am not sure how to proceed.

$2)$ Calculate $E(N_1|N_2)$:

First find conditional distribution of $N_1$ given $N_2 = n$

$$E(N_1\mid N_2 = n) = g(n)$$

$$E(N_1\mid N_2) = g(N_2)$$

But I get stuck here as well and not sure how to proceed.

Answer 1

1. Notice that $N_2-N_1\sim \text{Pois}(\lambda(2-1) = \lambda)$. Hence $$E[N_2-N_1] = \lambda(1) = \lambda.$$

2. You should know by now, or recognize that $N_1|N_2=n$ follows a binomial distribution $$\text{Bin}(n, p = 1/2).$$ You can show this by finding $$P(N_1 = k|N_2 = n)$$ using Bayes' rule. Hence $$E[N_1|N_2] = \frac12N_2.$$