Expectation of a product of 2 poisson process

Question

I'm trying to help a friend but we are stuck on a question. I hope you can help us !

We are trying to compute $E(N_1N_2)$ knowing that $N_t$ is following a Poisson process with $\lambda = 2$

Could you help us ?

We think that there is 2 cases. One where the two variables are independent which is quite easy but in the other case we are not sure of how we are supposed to do it.

Thanks in advance !

Answer 1

The two variables here are not independent, so it's not even a separate case. The key fact to remember is that a Poisson process has independent and stationary increments.

That is, we know that $N_1$ and $N_2-N_1$ are independent. Using this, we can write $$E[N_1N_2] = E[N_1(N_2-N_1+N_1)] = E[N_1(N_2-N_1)+N_1^2]$$

Can you continue?