40. (i) Let X, X' and Y, Y' be independent samples of size 2 from continuous distributions F and G respectively. Then p = P{max(X, X') < min(Y, Y')) + P(max(Y, Y') < min( X, X')} = + 24, where A (FG) d[(F+ G)/2]. (ii) A=0 if and only if F = G.

[(i): p (1F) dG + (1 - G) dF2, which after some computation re- duces to the stated form. - (ii): A0 implies F(x) = G(x) except on a set N which has measure zero both under F and G. Suppose that G(x) F(x)=n>0. Then there exists x such that G(x) = F(x) + n and F(x) < G(x) for x x x. Since G(x) G(x) > 0, it follows that A > 0.] -