We now calculate the mean and variance of the estimator y. Since the sample mean is a linear function of the observations, its expectation is simple to calculate as follows:
E y E
( ) = ⎛
∑ ∑1 1 1 1
This shows that the expected value of the least-squares estimator (the sample mean) is equal to the projection coefficient (the population mean). An estimator with the property that its expectation is equal to the parameter it is estimating is called unbiased.