Buffon's Needle problem

Question

So I'm working through some computational stats stuff from a free pdf of a book. Specifically I'm looking at their take on the classic Buffon's needle problem. The question has a theoretical part and a computational part. My theoretical background isn't very strong, so I did some research to get those questions answered for me, so I can better understand the computational problem. And the theory seems fairly straight forward on explanation.

My issue is the precursor question to the actual computation problem. The question is this: Let T be the number of crossings in n tosses of the needle, then E=Td/(nl)is an unbiased estimator of 2/π. Calculate the variance of E and thus suggest the best needle length l to use, subject to the restriction l ≤ d.

How would I calculate this? And the best l is simply going to be the one that minimizes the variance of the estimate, right? I understand the definition of variance, but I have no idea how to apply them here.

Answer 1

For simplicity I will call your estimator $\theta$

$$\begin{align} \operatorname{var}(\theta) &= \operatorname{var}{[\frac{Td}{nl}}]\\ &= \frac{d^2}{n^2 l^2} \operatorname{var}[T]\\ \end{align}$$

The event of a needle crossing can just be considered a single trial in a bernoulli experiment with $P = \frac{2l}{d \pi}$. Therefore $T$ is binomially distributed with $n=n$, $p = P$.

The variance, $\operatorname{var}[T]$, then just follows: $np*(1-p)$.

Differentiate with respect $l$ then solve.