Statistical estimation of the number π by rectangles in Buffon’s needle problem

The Buffon’s needle problem is a random experiment leading to estimate of the number π by ”randomly” throwing a needle onto a plane partitioned by parallel lines. Indeed, in the independently repetitions of the experiment, based on the number of times where the needle will cross a line, one can construct an estimator of π. The aim of this note is to obtain a better estimator (in some sense) by considering a model where the plane is partitioned by rectangles. We show that both estimators are asymptotically normal and unbiased; and also the confidence intervals are obtained for π. We calculate the asymptotic relative efficiency of the estimators and show that the estimator based on the rectangles is more efficient. The data of a real experiment is provided.