(a) Suppose X1, X2,...,Xn are IID N (, 1). Find the mean and variance of the MLE...

(a) Suppose X1, X2,...,Xn are IID N (µ, 1). Find the mean and variance of the MLE for µ. Find the distribution of the MLE andcompare to the theorem. Show that −L

n(µ)/n ˆ → Iµ. Comment: for the normal, the asymptotics are awfully good.

(b) If U is N (0, 1), show that U is symmetric: namely, P (U < y) = P (−U < y). Hints. (i) P (−U < y) = P (U > −y), and (ii) exp(−x2/2) is a symmetric function of x.