# 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 will be an unbiased estimator if the expected value of equals the true value of the parameter

i.e.

Consider the following worked 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, we have

Since are each random variable, their expected value will be equal to the probability mean,

Therefore,

Hence is an unbiased estimator of the population mean.