Unbiasedness of an Estimator

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

i.e.

Consider the following worked example

Example:

Show that the sample mean $\overline X$ is an unbiased estimator of the population mean$\mu$.

Solution:

In order to show that $\overline X$ 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 ${X_1},{X_2},{X_3}, \ldots ,{X_n}$ are each random variable, their expected value will be equal to the probability mean$\mu$,

Therefore, $E\left( {\overline X } \right) = \mu$

Hence $\overline X$ is an unbiased estimator of the population mean$\mu$.