# Euclidean distance for points in $\mathbb R^2$

I have a point, call it $x$, located somewhere in a unit square $[0,1]^2$.

I drawn $n$ new points, all uniformly and independently, all those in the unit square $[0,1]^2$.

What is the expected Euclidean distance between $x$, and the closest of the other $n$ dots?

EDIT: thanks all for your nice replies. I do realize it is incredibly difficult. If the $n$ numbers are drawned from a Poisson point processes, a paper by Holroyd et al. (Poisson point proccesses) shows that the probability that two points are separated by a distance larger than r is

$Pr(Q>r)=\frac{C}{r^{0.49}}$

I was trying to derive this result limiting myself to points in the unit square but it looks hard.

• @Josué Ortega This video by MindYourDecisions on YouTube (youtube.com/watch?v=i4VqXRRXi68) uses integrals to find the average distance between two points in the unit square - it's extremely relevant to your question. – Toby Mak Jul 4 '17 at 12:13
• This looks like a problem where the sane response would be to throw up your hands and do a Monte Carlo simulation. – Henning Makholm Jul 4 '17 at 12:29