7.4.8 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 = Rcos(2~U), Y = Rsin(2rU). (7.28)

Find: (i) The joint density of ( X , Y ) . (ii) P{XY > 0). (iii) P { X > 0). (iv)

P { X 2 + Y 2 5 t } . [Hint: For (i) find the joint density of (R, U ) first and then use transformation (7.28).]