Problem 5.62 Let (X, Y ) be uniformly distributed over the unit circle {(x, y) : x2 + y2 ≤ 1}. Its joint distribution function is given by f(x, y) = 1 π , for x2 + y2 ≤ 1 f(x, y)=0, elsewhere. Compute:

(a) P(X2 + Y 2 ≤ 1/4).

(b) P(X>Y ).

(c) P(X = Y ).

(d) P(Y < 2X).

(e) Let R = X2 + Y 2. Compute FR(r) = P(R ≤ r). Hint: No calculations involving double integrals are necessary. Just sketch the region of integration and evaluate the probability by first computing the corresponding area.