Consider estimating the mean of a variable drawn from a population with mean u and standard deviation o, using the following formulae: Ĥ1 = x1. X + x2 2 X1 + 2x2 X +x2 3 is = %3D Assume that in all cases the individual estimates x, are independent of each other. Determine the bias, variance, and mean square error for each of the estimators. Which ones are biased?

We define the bias of an estimator H as the expected value of the estimator less the value $\theta$ being estimated.

Bias = E(H) - $\theta$

For our case ,the bias is defined as below

Bias = E (x1. X + x2 2 X1 + 2x2 X +x2 ) - (u)

variance = E( H - E(H) )2

= E( (x1. X + x2 2 X1 + 2x2 X +x2 ) - E(x1. X + x2 2 X1 + 2x2 X +x2 ) )2

Mean squared Error = variance + ( Bias)2

Thus we substitute the variance and bias obtained above to get the mean squared error.