4. The joint cumulative distribution function, or joint CDF, of the random variables X and Y is defined as F(x, y) = P(X ≤ x,Y ≤ y). Let X and Y be the random variables of Exercise 1.

(a) Make a table for the F(x, y) at the possible (x, y)

values that (X,Y) takes.

(b) The marginal CDFs of X and Y can be obtained from their joint CDF as FX(x) = F(x,∞), and FY(y) = F(∞, y). Use these formulas to find the marginal CDFs of X and Y.

(c) It can be shown that the joint PMF can be obtained from the joint CDF as P(X = x,Y = y) = F(x, y) − F(x, y − 1)

−F(x − 1, y) + F(x − 1, y − 1).

(This is more complicated than the formula P(X = x) = FX(x) − FX(x − 1) for the univariate case!) Use this formula to compute P(X = 2,Y = 2), and confirm your answer from the PMF given in Exercise 1.