where are you stuck and what did you try?
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).