Let (X, Y ) be uniformly distributed in a circle of radius r about the origin. That is, their joint density is given by

Let R = √
X2 + Y2 and θ = arc tanY/X denote their polar coordinates. Show that R and θ are independent with θ being uniform on (0, 2π) and P{R

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1 f(x, y) = 2; 0 x + y r r2