7. (Total: 14 points) Suppose that a fair six-sided dice with faces numbered 1 to 6 is thrown twice and the two numbers shown are denoted by U and V. Denote X = max(U, V) and Y = min(U, V). (a) (2 points) Write down the cumulative distribution function of U. (b) (1 point) Find Pr(X = Y). (c) (2 points) Explain whether X and Y are independent. (d) (i) (2 points) Show that the cumulative distribution function of X is 0, for x < 1; Fx(x)= for 1 x 6; 36 1, for x > 6. (ii) (2 points) Find an algebraic form of the probability mass function of X. (e) (3 points) Show that the probability mass function of Y is PY (y) 13-2y 36 " for y 1, 2, 3, 4, 5, 6; otherwise. (f) (2 points) Find the joint probability mass function of X and Y.