# The probability distribution for the number of points in a randomly sampled area of a plane covered with $N$ points (placed with uniform probability)

With uniform probability, I place $N$ points on a plane of dimensions $D_x \times D_y$ and sample a region of area $A \leq D_x*D_y$ which has arbitrary geometry. What is the probability distribution for the number of points in this area? Also, as a function of the area of the region I sample, how accurately can I guess the total number of points on the bounded plane?

## 1 Answer

Imagine that the points were placed one at a time, independently of course. Let $P_1,P_2,\dots,P_N$ be the points, listed in the order they were placed. The probability that $P_i$ is in our set is $\frac{A}{D_xD_y}$. Call this $p$.

We want the probability that exactly $k$ of the points are in our set. The distribution is binomial, and the required probability is $\binom{N}{k}p^k(1-p)^{N-k}$.

The obvious estimator of $N$ is $\frac{D_xD_y}{A}X$. We have not worked out a measure (with proof) of its reliability.

• How would we estimate a confidence interval for $N$ provided the number of points in the sampled area? – user71678 Apr 9 '13 at 5:27
• I have not done the calculation. A crude guess is that we can se the estimator as a good enough guess of $N$ for the purpose of estimating the variance of $X$. Use that to estimate the variance of our estimator, and crossing one's fingers, hope that the estimator is close enough to normal. That gives us a confidence interval, one that I would feel moderately comfortable using at least for ballpark estimates when the ratio of $A$ to total is not too small. However, my derivation is too informal for an official answer. – André Nicolas Apr 9 '13 at 5:37
• Thanks a lot for your answer - it's very helpful. However, I'd love to see a proof! – user71678 Apr 9 '13 at 5:42