1. Suppose a sample X_{1}, X_{2},…X_{n}is drawn randomly from a normal distribution N(m,s^{2}).

a) Show that Xbar (average) and X_{i}-Xbar are uncorrelated for i=1,2,...n;

b) Show that Cov(Xbar, s^{2}) = 0;

c) Show that m'=X^{2}bar - s^{2}/n is unbiased estimator of m^{2}, where X^{2}bar is the average of X^{2}.

2. For uniform distribution E~U[2m_{1}m_{2};m_{1}^{2}+m_{2}^{2}], construct estimates for m_{1}and m_{2}using:

a) method of moments;

b) method of maximum likelihood.

(m_{2}>m_{1}>0).