# Unbiasedness of an Estimator

This is probably the most important property that a good estimator should possess. According to this property, if the statistic is an estimator of , it will be an unbiased estimator if the expected value of equals the true value of the parameter

i.e.

Consider the following working example.

__Example__**:**

Show that the sample mean is an unbiased estimator of the population mean.

__Solution__**:**

In order to show that is an unbiased estimator, we need to prove that

We have

Therefore,

From the rule of expectation, the expected value of a linear combination is equal to the linear combination of their expectations. So we have:

Since are each random variables, their expected values will be equal to the probability mean ,

Therefore, .

Hence is an unbiased estimator of the population mean .