A box contains three cards, labeled 1, 2, and 3. Two cards are chosen at random, with the first card being replaced before the second card is drawn. Let X represent the number on the first card, and let Y represent the number on the second card.

Question

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).

Answer 1

The possible values of the pair (X, Y) are the ordered pairs (x, y) where each of x and y is equal to 1, 2, or 3. There are nine such ordered pairs, and each is equally likely. Therefore (x, y) = 1/9 for x= 1, 2, 3 and y= 1, 2, 3, and (x, y) = 0 for other values of (x, y).



(b)Both X and Y are sampled from the numbers {1, 2, 3}, with each number being equally likely. Therefore (1) = pX(2) = (3) = 1/3, and (x) = 0 for other values of x. is the same.



(c) =

= 1(1/3) + 2(1/3) + 3(1/3) = 2



(d)

Another way to compute is to note that X and Y are independent, so == (2)(2) = 4.



(e) Cov(X, Y) = - = 4 – (2)(2) = 0