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
Consider the following working example.
Show that the sample mean is an unbiased estimator of the population mean.
In order to show that is an unbiased estimator, we need to prove that
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 ,
Hence is an unbiased estimator of the population mean .