18.11 Suppose that X and Y are distributed bivariate normal with the density given in Equation (18.38).

a. Show that the density of Y given X = x can be written as fYX= x(y) =

1 sYX22p expc -

1 2

a y - mYX sYX b

2 d

where sYX = 2s2 Y(1 - r2 XY) and mYX = mY + (sXYs2 X)(x - mX).
[Hint: Use the definition of the conditional probability density fYX= x(y) = gX, Y (x, y)>fX (x), where gX,Y is the joint density of X and Y and fX is the marginal density of X.]

b. Use the result in

(a) to show that Y X = x ~ N(mYX, s2 YX).

c. Use the result in

(b) to show that E(Y X = x) = a + bx for suitably chosen constants a and b.