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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?

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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 $\lim_{n\to\infty}P(X_n=k)=e^{-c}\frac{c^k}{k!}$.

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  • $\begingroup$ Yes. Stirlings formula is a tool to prove that. Looks okay to me. $\endgroup$ – drhab May 18 '18 at 8:03
  • $\begingroup$ How is Stirlings formula applicable? oxfordmathcenter.com/drupal7/node/297 I pretty much followed this idea and it seems like you can do it without needing an approximation for n! I mean, I guess it doesn't really matter if I HAVE to use it, but it makes me feel like I'm missing something $\endgroup$ – Brandon Barry May 18 '18 at 8:04
  • $\begingroup$ Sorry, but I will not dive further into Stirlings formula. Especially the fact that it is not even necessary to make use of it makes me reluctant. I prefer enjoying that we can do without. $\endgroup$ – drhab May 18 '18 at 8:20
  • $\begingroup$ No worries. Also, with the part about "the number of points that fall within the unit circle" does that affect our answer? I'm assuming that the r = sqrt(n/c) comes into play here, so with r=1 doesn't this just make n=c and then the distribution will be the same but with n instead? $\endgroup$ – Brandon Barry May 18 '18 at 8:59

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