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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 $N_1^2$.

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  • $\begingroup$ 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... $\endgroup$
    – Surb
    Sep 13, 2021 at 16:30

1 Answer 1

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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).

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