Suppose that the two-dimensional vectors (X1, Y1), (X2, Y2), . . . , (Xn, Yn) form a random sample from a bivariate normal distribution for which the means of X and Y, the variances of X and Y , and the correlation between X and Y are unknown. Shows that the M.L.E.€™s of these five parameters are as follows:

First, rewrite the joint p.d.f. of each pair (Xi, Yi) as the product of the marginal p.d.f. of Xi and the conditional p.d.f. of Yi given Xi. Second, transform the parameters to μ1, σ21 and

Third, maximize the likelihood function as a function of the new parameters. Finally, apply the invariance property of M.L.E.€™s to find the M.L.E.€™s of the original parameters. The above transformation greatly simplifies the maximization of the likelihood.

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(Xi_XyY i=1 and ơi rI i=1 ρσ2