# Expected value and Poisson Process

I have an assignment, but I have some problems doing it.

Let $$\{N(t); t>0\}$$ be a Poisson Process with intensity $$\lambda$$ > $$0$$. Compute $$E(N_1^2N_2)$$.

I have tried something.

\begin{align} E(N_1^2N_2) &= E((N_2-N_1)(N_1-N_0)^2) + E(N_2^2) \\ &=[E(N_2)-E(N_1)]E(N_1^2) + E(N_2^2)\end{align}

From the first to the second lign, I'm using the fact that $$(0, 1]$$ and $$(1, 2]$$ are non overlapping intervals (which implies that $$(N_2-N_1)$$ and $$(N_1-N_0)$$ are independent) and that $$N_0=0$$.

The main problem is that I'm really not sure of this because of the fact that we consider here $$N_1^2$$.

• Your calculation is strange : it should be $$\mathbb E[N_1^2N_2]=\mathbb E[N_1^2(N_2-N_1)]+\mathbb E[N_1^3],$$ and use the fact that $N_2-N_1$ and $N_1$ are independent...
– Surb
Sep 13, 2021 at 16:30

Your idea of looking at disjoint intervals is good, but you have a minor typo. Note that the first line should be $$E[N_1^2 N_2] = E[(N_2-N_1)N_1^2] + E[N_1^3].$$
Next, your claim that $$E[(N_2-N_1)N_1^2] = E[N_2-N_1]E[N_1^2]$$ is also correct. If $$X$$ and $$Y$$ are independent, then $$E[f(X)g(Y)] = E[f(X)]E[g(Y)]$$ for any $$f, g$$ (such that the expectations exist).