Find limiting distribution of points on a circle?

Question

Say there are n points uniformly distributed over the surface of a circular disk of radius r=sqrt(n/c) where c is a positive constant. Find the limiting distribution as n -> ∞ of the number of points that fall within the unit circle. I'm honestly so stuck on this, so any hints would be appreciated. All I can guess at the moment is to use a uniformly distributed r [0,1] and θ [0,2π] and see if I can find the joint PDF, then take the limiting distribution of that?

Answer 1

Hints:

• For a fixed $n$ you are dealing with binomial distribution having parameters $n$ and $p=cn^{-1}$.

• If $X_n$ has that distribution for $n=1,2,\dots$ then for fixed $k$ it can be shown that $\lim_{n\to\infty}P(X_n=k)=e^{-c}\frac{c^k}{k!}$.