Calculating a certain probability in a Poisson process

Question

Let $N(\cdot)$ be a Poisson process with intensity $\mu$. I want to compute the following probability:

$$ P(\{N((1,3]) > 0\}\cap \{N((2,4]) > 0\}) $$

I tried to split the involved intervals in half so as to achieve some sort of independence between the involved random variables, but I did not succeed. I also tried computing

$$ P(\{N((1,3]) = 0\}\cup \{N((2,4]) = 0\}) $$

but that did not yield any helpful results either. Any suggestions on how to approach this would be greatly appreciated.

Answer 1

Well: $$\mathsf P((A\cup B)\cap(B\cup C)) = \mathsf P(B)+\mathsf P(A\cap B^\complement\cap C)$$

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And $(N_{(1;3]}{>}0)\cap( N_{(2;4]}{>}0)$ if and only if $(N_{(0;2]}{>}0\cup N_{(2;3]}{>}0)\cap (N_{(2;3]}{>}0\cup N_{(3;4]}{>}0)$.

Therefore:

$$\mathsf P(N_{(1;3]}{>}0\cap N_{(2;4]}{>}0)~=~\mathsf P(N_{(2;3]}{>}0)+\mathsf P(N_{(1;2]}{>}0)\,\mathsf P(N_{(2;3]}{=}0)\,\mathsf P(N_{(3;4]}{>}0)$$