Expectation of the product of two dependent Poisson process

Question

Consider $0<a<b<c<d$. Let $N(t)$ be number of count up to time t.

Let $X := N(d)-N(b)$ and $Y:=N(c)-N(a)$

Find $\mathbb{E}[XY]$

Since $X,Y$ no longer independent, we have $E(XY) \ne E(X)E(Y)$.

What I'm thinking try to create a random variable $Z$ that independent of one of them but end up with creating more dependent random variable and can't break up the product within the expectaiton. Any idea?

Answer 1

Guide:

$$X = [N(d)-N(c)]+[N(c) - N(b)]$$

$$Y = [N(c)-N(b)]+[N(b)-N(a)]$$

Note that we are able to compute $E[((N(c)-N(b))^2]$ as we know the distribution of $N(c)-N(b)$.