Let $D$ be a Poisson Point Process on $\mathbb{R}$ of rate $\lambda > 0$. Suppose $(X_t)_{t \geq 0}$ is a continuous time Markov Chain taking values $\{0,1\}$. The chain and the point process are independent. Now define:
$Z = \int_0^1\ 1_{\{X_t = 1\}} dt $
I wonder if it's true that:
$|D \cap \{ t :X_t = 1, t \in [0,1]\}| \sim |D \cap [0,Z]| $
I guess it is true and one way I tried to approach this problem is to condition both events $\{|D \cap \{ t :X_t = 1, t \in [0,1]\}| = k \}$ and $\{|D \cap [0,Z]| = k \}$ on the jumping times $\{\tau_i\}_{i \geq 0}$ of the chain. But I can show the equality only conditioned on the times assuming specific values, i.e. $\{\tau_i=r_i \; \; \forall i\geq0\}$ for some $\{r_i\} \in \mathbb{R}$, which I guess it's not enough. Any thoughts on how I can go from this to prove claim?