Final answer:
a. The joint support is the set of all possible pairs (x, y) that can be formed by selecting an even integer from one set and an integer from another set. b. The marginal pmf of x is the probability distribution of x alone, without considering y. c. X and y are independent.
Step-by-step explanation:
a. The joint support is the set of all possible pairs (x, y) that can be formed by selecting an even integer from the set {0, 2, 4, 6, 8} and an integer from the set {0, 1, 2, 3, 4}.
To find the joint pmf, we need to determine the probability of each pair (x, y) occurring. Since x and y are chosen randomly, each pair has an equal probability of occurring. Therefore, the joint pmf is:
P(x, y) = 1/5 * 1/5 = 1/25 for all pairs (x, y).
b. The marginal pmf of x is the probability distribution of x alone, without considering y. Since the probability of selecting any even integer from the set {0, 2, 4, 6, 8} is 1/5, the marginal pmf of x is:
P(x) = 1/5 for x = 0, 2, 4, 6, 8.
The marginal pmf of y is the probability distribution of y alone, without considering x. Since the probability of selecting any integer from the set {0, 1, 2, 3, 4} is 1/5, the marginal pmf of y is:
P(y) = 1/5 for y = 0, 1, 2, 3, 4.
c. To determine if x and y are independent, we need to check if the joint pmf is equal to the product of the marginal pmfs. In this case, P(x, y) = P(x) * P(y) for all pairs (x, y), so x and y are independent.