28. The Hit–Miss Method: Suppose g is bounded in [0, 1]—for instance, suppose 0 g(x)b for x ∈ [0, 1]. Let U1, U2 be independent random numbers and set X = U1,Y = bU2—so the point (X,Y) is uniformly distributed in a rectangle of length 1 and height

b. Now set

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

(a) Show that E[bI] = 1 0 g(x) dx.

(b) Show that Var(bI) Var(g(U)), and so hit–miss has larger variance than simply computing g of a random number.

Transcribed Image Text:

[1, I = if Y