Suppose we model a dartboard as the unit disk in the plane: D = {(x, y) : x^2 + y^2 1}. A dart is thrown at this board in such a way that every point in D is equally likely to be hit. Let (X, Y) represent the (random) point where the dart lands. Using the joint pdf f(x, y) = 1/ for (x, y) D answer the following. (a) Derive the marginal pdf's of X and Y . Clearly label each. Are the random variables X and Y independent? Justify your assertion.

(b) Compute P(X 1 2 |Y 0). Note: this is just an ordinary conditional probability since the conditioning event has positive probability.

(c) Suppose the dartboard were a rectangle instead, say, D = {(x, y) : a x b, c y d} with a < b and c < d. If each point in D is equally likely write down the joint pdf of (X, Y ).

(d) (continued from (c)) Derive the marginal pdf's of X and Y . Clearly label each. Are the random variables X and Y independent? Justify your assertion.