8.7 In this problem we refer to Example 8.6. Suppose the random variables

(X1, Y1),(X2, Y2),...,(Xn, Yn) are all independent and that for each i we have Xi, Yi distributed as N(μi, σ2).

a) Set ν = 1

σ2 and express the likelihood function in terms of the parameters

μ1,...,μn, and ν.

b) Show that the log likelihood is l(μ1,...,μn, ν)=2n log 1

√2π + n log ν − ν

n i=1

(Xi − μi)2 + (Yi − μi)2 2 .

c) Calculate the MLEs for the parameters μ1,...,μn, ν.

d) Derive that the MLE for σ2 is σˆ2 = 1 2n n

i=1 s2 i , where s2 i = (Xi − μˆi)

2 + (Yi − μˆi)

2.

e) Show that the statistics s2 i are i.i.d. and calculate their expected value.

f) Use the law of large numbers to show that