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

for −∞ 0,σy > 0,−∞

(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) = 1 - 2 1 - x-lx exp{-2(1-p) 2p(x-x)(y-ply) + xy y - My Oy