Originally Posted by

**SpringFan25** suppose you have 1 realisation of x and your estimator is c(x)

For your estimator to be unbiased you require:

$\displaystyle E(c(x)) = p$

$\displaystyle \sum c(x) \times p(1-p)^{x-1} =p$

where the summation is taken over all possible values of x (1,2,3,4,5,6...)

Now, consider the following function:

c(x)=1 if x=1

c(x)=0 otherwise

The expected value of our function is then

$\displaystyle \sum c(x) \times p(1-p)^x =p$

$\displaystyle =(f(1)\times p) + (f(2)\times p(1-p)) + (f(3)\times p(1-p)^2) +... $

$\displaystyle =(1 \times p) + (0 \times p(1-p)) + (0 \times p(1-p)^2) +... $

$\displaystyle =p$

as required

So your estimator is $\displaystyle c(x)$ where c(x) was defined above.

In reality, this estimator is no practical use. But it is an unbiased which is all the question asked for. I did assume that your sample is a single realisation from the distribution. if you have a sample with multiple data points, just discard all but one of them and the proceedure still works.