16. The random variables X and Y are said to have a bivariate normal distribution if their joint density function is given by

(a) Show that X is normally distributed with mean μx and variance σ2 x , and Y is normally distributed with mean μy and variance σ2 y .

(b) Show that the conditional density of X given that Y = y is normal with mean μx + (ρσx/σy)(y − μy) and variance σ2 x (1 − ρ2).
The quantity ρ is called the correlation between X and Y. It can be shown that

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f(x,y): 2 exp { 2(1-p) 2 x-fx y-py + X Ox Oxy for-00 < x <00, -00 < y < oo, where x, y, x, y, and p are constants such that -1 0, oy > 0, -00 < x <00, -00