18.7 Suppose that X and u are continuous random variables and (Xi, ui), i = 1,

c, n, are i.i.d.

a. Show that the joint probability density function (p.d.f.) of (ui, uj, Xi, Xj)

can be written as f(ui, Xi) f(uj, Xj) for i  j, where f(ui, Xi) is the joint p.d.f. of ui and Xi.

b. Show that E(uiuj  Xi, Xj) = E(ui  Xi) E(uj  Xj) for i  j.

c. Show that E(ui  X1,

c, Xn) = E(ui  Xi).

d. Show that E(uiuj  X1, X2,

c, Xn) = E(ui  Xi) E(uj  Xj) for i  j.