Let $C$ be a unit circle and $P$ be a Poisson point process on $C$ with intensity $1$. For each point $p$ of the process we draw an interval $(p-\epsilon, p+\epsilon)$ for some $\epsilon>0$. Further let's denote by $L$ Lebesgue's measure of the union of the intervals.

What is the **law** of $L$?

What can be done easily is for example calculating $\mathbb{E}L$. Namely, we have

$$L=\int_{C} 1_{(x-\epsilon, x+\epsilon)\cap P\neq \emptyset}\text{d}x,$$ which gives $$\mathbb{E}L=\int_{C} \mathbb{P}((x-\epsilon, x+\epsilon)\cap P\neq \emptyset)\text{d}x=(2\pi)\mathbb{P}((-\epsilon, +\epsilon)\cap P\neq \emptyset) = (2\pi)(1-\exp(-2\epsilon)).$$

In principle this method can be use to calculate recursively moments of arbitrary order but it seems very cumbersome.