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 0 X2 Y2, tan1 Y X denote the polar coordinates Compute the joint density of R, and use this to give a simulation method Another meth...

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4. Suppose we want to simulate a point located at random in a circle of radius r...

Question:

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 = 0 X2 + Y2, θ = tan−1 Y/X denote the polar coordinates. Compute the joint density of R, θ and use this to give a simulation method. Another method for simulating X,Y is as follows:
Step 1: Generate independent random numbers U1,U2 and set Z1 = 2rU1− r, Z2 = 2rU2−r. Then Z1,Z2 is uniform in the square whose sides are of length 2r and which encloses, the circle of radius r (see Figure 11.6).
Step 2: If (Z1,Z2) lies in the circle of radius r—that is, if Z2 1 + Z2 2 r2—set (X, Y) = (Z1,Z2). Otherwise return to step 1.

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

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