only 4.2 is needed

PROBLEM # 4: Suppose we observe 3 sets of independent random variables: X1, X2, ... .Xn ~ N(Ox, 02); Y1, Y2, . .. , Yn ~ N(Oy, 02); and Z1, Z2, . ..,Zn ~N(Or + 0y, 02), which are also independent of each other. Let S2 = En (Xi-X)?/(n-1), $2 = En,(Yi-Y)?/(n-1),and S? = Chi(Zi -Z)2/(n-1), where X, Y and Z are respectively the sample means for the X's, Y's and Z's. Also we let S? = (1/3) ($2 + S2 + S?). 4.1 Use all observations to obtain the MLE of o2. 4.2 What is the asymptotic distribution of the MLE, oh? If you have a problem to find the answer, find the asymptotic distribution of $2 = (1/3) (S2 + S2 + S2) to obtain maximum of 4 points instead (you do not need to do this if you obtain the answer for the MLE)