A target is a disk of radius 3 feet centered at the origin. A competitor fires a

Question

A target is a disk of radius 3 feet centered at the origin. A competitor fires a projectile at the target and receives 30 points if it lands within 1 foot of the center, 20 points if it lands between 1 and 2 feet of the center, and 10 points if it lands between 2 and 3 feet of the center (and 0 points otherwise). Let (X,Y) represent the coordinates where the projectile hits and let S be the score.
2. Suppose the position of the projectile on the target is a bivariate normal with X and Y independent standard normal with 'rho' = 0.
(a) Write the joint PDF for (X,Y)
(b) Find the joint pdf of (R, 'theta'), the polar coordinates of the point where:
R = sqrt(X^2 + Y^2) and 'Theta' = arctan(Y/X)
(c) Are R and 'theta' independent?

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