If your sample is , the unbiased estimate of . So a guess for the unbiased estimate of would be .

Recall that where is the probability density function, in this case . Therefore,

But if we want the estimator to be unbiased, we should have

Therefore, we see that our initial guess should be modified to be

This is the unbiased estimator of .

In general this is a trick you can use to find an unbiased estimator. Make a good guess that will make , then take .