Finding the conditional expectation of Poisson Process, $E(N_2\mid N_1)$

Question

Not quite sure how to proceed here in finding $E(N_2\mid N_1)$ where $N_t$ is a poisson process with $\lambda=1$.

Can't use the law of total expectation or Bayes (since it's expectation?) Am I meant to decompose it into an integrand? i.e. $$\int \limits_{n=k}^\infty P(N_2=k\mid N_1=n)P(N_1=n)?$$

Answer 1

We have \begin{align} E[N_2|N_1] &= E[N_2-N_1|N_1]+E[N_1|N_1]\\ &=\lambda (2-1)+N_1 \end{align}