Calls arrives according to a Poisson arrival process with rate $\lambda = 15$. Find $E(N(2,4]N(3,5])$
My thoughts: $E(N(2,4]) = E(N(3,5]) = \lambda * t = 15 * 2 = 30$ However, I cannot figure out the next step, since $(2,4]$ and $(3,5]$ are NOT non-overlapping, so they are NOT independent and I cannot get the resulting expectation by simply multiplying them.
I'm really stuck in this. Can someone please explain this problem? Many thanks.
Let $X$ be the number of arrivals between $2$ and $3$, $Y$ the number between $3$ and $4$, and $Z$ the number between $4$ and $5$.
We want $E((X+Y)(Y+Z))$. Expand, and use the linearity of expectation. Almost everything is simple, except that we need $E(Y^2)$. You can write this down easily if you already know the variance of the Poisson, since $\text{Var}(Y)=E(Y^2)-(E(Y))^2$.