4. Suppose we want to simulate a point located at random in a circle of radius r centered at the origin. That is, we want to simulate X, Y having joint density

(a) Let R = \lx2 + Õ2, è = tan"1 Y/X denote the polar coordinates.

Compute the joint density of R, è and use this to give a simulation method.

Another method for simulating X, Y is as follows:

Stepl: Generate independent random numbers Ul9 U2 and set Zx - 2rUx - r, Z2 = 2rU2 - r. Then Zx, Z2 is uniform in the square ö whose sides are of length 2r and which encloses the circle of radius r (see Figure 11.6).

Step 2: If (Zj, Z2) lies in the circle of radius r—that is, if Z\ + Z\ < r2

—set (X, Y) = (Zj, Z2). Otherwise return to Step 1.

(b) Prove that this method works, and compute the distribution of the number of random numbers it requires.