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

ç * 1 Ä*,y) = - / exp à 1 2 ð ó , ó ^ í à í C 2(1 - p2)

x - ì÷\2 _ 2p(x - px)(y - ìã) (y - ìã

÷ô

for - o o < ÷ < oo, - o o < y < oo, where ó÷, oy, ì÷, ìã, and ñ are constants such that - 1 < ñ < 1, ó÷ > Ï, ay > Ï, - o o < ì÷ < oo, - o o < ìã < <÷>.

(a) Show that Ë" is normally distributed with mean ì÷ and variance ó2., and Y is normally distributed with mean ìã and variance ó^.

(b) Show that the conditional density of X given that Y = y is normal with mean ì÷ + (pcx/ay)(y - ìã) and variance ó2(1 - ñ2).

The quantity /? is called the correlation between X and Y. It can be shown that E[(X - ì÷)(Õ - My)]

Ñ = —

axGy _ Cov^, Y)

axoy