# Probability density function of distance between two points

There is a square whose vertices are at $(x,y)=(0,0),(0,1),(1,0),(1,1)$, and Roger is located at point $(0,1/2)$. There are $N$ trees distributed randomly and independently over the unit square. Also in the whole $x,y$ plane, stones are distributed independently according to a two-dimensional Poisson point process. If $A$ is the random variable denoting the distance between Roger and the closest stone to a random tree, what is the probability density function (pdf) of $A$, $f(x)$?

The brute-force suggests to find the pdf of the distance between the closest stone to a random tree and then find the pdf of distance between a random tree and Roger and then use multiple integrals to solve it. Any simpler method?

Particularly, that would be helpful to show $f(x)\leq 2 \pi x$ and $P(A\leq x)\leq \pi x^2$.

• Tags stochastic-processes (where do you see one?) and stochastic-calculus (where do you see some?) both inappropriate here, please read the tag info before using a tag. – Did Aug 19 '17 at 10:38