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 Fig. 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 0, if Y