28. The Hit-Miss Method: Suppose g is bounded in [0, 1]—for instance, suppose 0 < g(x) < b for ÷ å [0, 1]. Let Ui9 U2 be independent random numbers and set X = Ux, Y = bU2—so the point (X, Y) is uniformly distributed in a rectangle of length 1 and height

b. Now set

/ = Ã1, ifY

^0, otherwise That is accept (X, Y) if it falls in the shaded area of Figure 11.7.

(a) Show that E[bl] = &g(x)dx.

(b) Show that Var(W) > Var(g((/)), and so hit-miss has larger variance than simply computing g of a random number.