Let (X1, X2) have a bivariate normal distribution with mean
vector (µ1, µ2), variance σ 12 for X1 and σ 2
2 for X2 and correlation cor(X1, X2) = ρ.
(a) Write down the joint density f(x1, x2).
(b) Find the marginal distribution f(x1)
(c) Find the conditional distribution f(x1 | x2) and the mean
and variance of the conditional distribution.
(d) Obtain the likelihood equations and calculate the MLE for
µ1, µ2, σ12 , σ2 2 , ρ.