Refer to Exercise 31. Assume the first card is not replaced before the second card is drawn.
a. Find the joint probability mass function of X and Y.
b. Find the marginal probability mass functions 
(x) and 
(y).
c. Find 
and 
.
d. Find 
.
e. Find Cov(X, Y).
(a) The values of X and Y must both be integers between 1 and 3 inclusive, and may not be equal. There are six possible values for the ordered pair (X, Y), specifically (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2). Each of these six ordered pairs is equally likely.
Therefore
(x, y) = 1/6 for (x, y) = (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2), and (x, y) = 0 for other values of (x, y).
(b) The value of X is chosen from the integers 1, 2, 3 with each integer being equally likely. Therefore 
(1) = (2) = (3) = 1/3. The marginal probability mass function 
is the same. To see this, compute
(c) 
= 
= 1(1/3) + 2(1/3) + 3(1/3) = 2
(d) 
(e) 
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