4. (20 pts) Suppose there are 4 cards, numbered 1, 2, 3, 4. You first pick...

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4. (20 pts) Suppose there are 4 cards, numbered 1, 2, 3, 4. You first pick a card unformly at random. Let this card be X € {1,...,4}. Then you pick another card unformly at random among the remaining 3 cards. Let this card be Y € {1,...,4} (since you pick the card without replacement, we must have X ‡ Y). (a) (3 pts) Find the joint probability mass function px,y (x, y). (b) (3 pts) Find the expectations E(X), E(Y). (c) (3 pts) Find the variances Var(X), Var(Y). (d) (4 pts) Are X, Y independent? Prove your assertion. (e) (3 pts) Let A be the event that "X is an odd number", and B be the event that “Y ≤ 2". Are the events A, B independent? Prove your assertion. (f) (4 pts) Let Z = X+Y. Find the probability mass function pz (z). Also, find the expectation E(Z). 4. (20 pts) Suppose there are 4 cards, numbered 1, 2, 3, 4. You first pick a card unformly at random. Let this card be X € {1,...,4}. Then you pick another card unformly at random among the remaining 3 cards. Let this card be Y € {1,...,4} (since you pick the card without replacement, we must have X ‡ Y). (a) (3 pts) Find the joint probability mass function px,y (x, y). (b) (3 pts) Find the expectations E(X), E(Y). (c) (3 pts) Find the variances Var(X), Var(Y). (d) (4 pts) Are X, Y independent? Prove your assertion. (e) (3 pts) Let A be the event that "X is an odd number", and B be the event that “Y ≤ 2". Are the events A, B independent? Prove your assertion. (f) (4 pts) Let Z = X+Y. Find the probability mass function pz (z). Also, find the expectation E(Z).

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