0
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ How would we estimate a confidence interval for $N$ provided the number of points in the sampled area? $\endgroup$ – user71678 Apr 9 '13 at 5:27
  • $\begingroup$ 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. $\endgroup$ – André Nicolas Apr 9 '13 at 5:37
  • $\begingroup$ Thanks a lot for your answer - it's very helpful. However, I'd love to see a proof! $\endgroup$ – user71678 Apr 9 '13 at 5:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.