Finding the estimator of π(1-π) from a random sample of n Bernoulli trials.

Question

A random sample of n independent Bernoulli trials with success probability π results in R successes. Derive an unbiased estimator of π (1 − π).

So, from what I understand (correct me if anything I say is wrong), R is a random variable that follows a binomial distribution. However, I am unsure about how to approach this question. Could someone please talk me through it?

Answer 1

We have $\frac1nX$ and $1-\frac1nX$ as unbiased estimators of $\pi$ and $1-\pi$ respectively.

This makes us wonder what we can "expect" from: $$Z:=\frac1nX\left(1-\frac1nX\right)$$

I am not saying that this random variable will do the job, but it cannot harm to find the expectation of $Z$ (and eventually adjust).

So you could start with giving that a try.