16. The random variables X and Y are said to have a bivariate normal distribution if their joint density function is given by f (x,y) = 1 2πσxσy 2

1 − ρ2 exp

⎧

⎨

⎩ − 1 2(1 − ρ2)

×

,x − μx

σx

2

− 2ρ(x − μx )(y − μy )

σxσy

+

y − μy

σy

2

-

for −∞ 0, σy > 0, −∞ < μx < ∞, −∞ < μy < ∞.

(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

ρ = E[(X − μx )(Y − μy )]

σxσy

= Cov(X,Y)

σxσy