Covariance of dependent Poisson-processes.

Question

I have a pretty easy-to understand task, but I am struggling to solve it.

We have $X$, which is $\mathrm{Poi}(1)$ point process.

We have $X_1$, which happens in the $[0,3]$ interval, and $X_2$, which is on $[2,4]$ interval. They are both point processes, with same $\lambda=1$.

Question is $\operatorname{Cov}(X_1,X_2)$.

My approach:

Since $X$ is a Poisson-distribution, with $\lambda=1$, $E(X_1)=3, E(X_2)=2$ from Poisson-distribution definiton.

However, $E(X_1 X_2)$ seems to be the tricky one, since they are not independent.

I tried to seperate $X_1$ to $[0,2]+[2,3]$, and $X_2$ to $[2,3]+[3,4]$, but no idea how to move forward.

Any help appreciated

Answer 1

You might find it easier to try $Y_1$ has a Poisson distribution with mean $2$, $Y_2$ has a Poisson distribution with mean $1$, $Y_3$ has a Poisson distribution with mean $1$, all independent. This makes $X_1=Y_1+Y_2$ and $X_2=Y_2+Y_3$

Then you want $\operatorname{Cov}(Y_1+Y_2,Y_2+Y_3)$, which is not difficult to calculate: it is $\operatorname{Var}(Y_2)=1$