Suppose you have a random variable X with distribution Geom(p), and you would like to estimate p.
Since E(x)=1/p, a natural approach to estimate p is to get a sample from X and then take the reciprocal 1/X.
Show that this is NOT an unbiased estimator for p.
Suppose you take n i.i.d. samples X1, X2,...,Xn, and you let Y be the fraction of these samples that equal 1. Show that Y is an unbiased estimator for p.