Statistics Biased Estimator Question?

Let x1, x2,..., xn be a random sample of size n from a population with mean u and variance sigma^2.





(a) Show that x(bar)^2 is a biased estimator for u^2 .


(b) Find the amount of bias in this estimator.


(c) What happens to the bias as the sample size n increases?





If you can answer even just one part of the question, it would be much appreciated. Thanks in advance!|||to calculate the bias you calculate the expectation of the difference





E( xbar^2 - sigma^2) = E( xbar^2 - E(x^2 - 2x*xbar + xbar^2) ) = E(xbar^2 - x^2 + 2x*xbar - xbar^2) = E(- x^2 + 2x*xbar) = E(x*(2xbar - x))





There is no reason that the last term would vanish, hence it is a biased estimator.