Let R be a nonnegative random variable with density f(r). Let (X, Y ) be a bivariate distribution obtained as follows: First, randomly choose a value of R, and then chose a value of U according to its U(0, 1) distribution. Now, put X = R cos(2πU), Y = R sin(2πU). (6.28)

Find: (i) The joint density of variables X and Y. (ii) P{XY > 0}. (iii) P{X > 0}.

(iv) P{X2 + Y 2 ≤ t}. [Hint: For (i), find the joint density of (R, U) first and then use transformation (6.28).]