Question

attance as a tunction of p. Does the general shape of the graph make sense? 4. Suppose that X and Y are bivariate normal. (a) Suppose that ρ > 0. For what values of y is E(X|Y = y] > E(X)? Explain in plain (b) Suppose that ρ < 0. For what values of y is E(XIY = y] > E(X)? Explain in plain (c) Suppose that ρ-: 0. For what values of y is EXlYs: y) > E(X)? Explain in plain (d) For what values of ρ is E(XIY-31 a decreasing function of y? Explain in plain language. language language. language.

Answer 1

$E(X|Y = y ) = \mu_X + \rho \frac{\sigma _X}{\sigma _Y}(Y - \mu _Y)$

a)

$E(X|Y = y ) - \mu_X = \rho \frac{\sigma _X}{\sigma _Y}(Y - \mu _Y)$

$E(X|Y = y ) - \mu_X > 0 \Rightarrow \rho \frac{\sigma _X}{\sigma _Y}(Y - \mu _Y) > 0$

$\Rightarrow \rho (Y - \mu _Y) > 0$

when $\rho$ > 0

hence

Y > $\mu _Y$

b)

when $\rho$ < 0

hence

Y < $\mu _Y$

c)

when $\rho$ = 0

it is not possible that E(X|Y = y) > E(X)

as E(X|Y = y)= > E(X)