Find the marginal probability mass functions pX (x) and pY (y).

1.  A penny and a nickel are tossed. The penny has probability 0.4 of coming up heads, and the nickel has probability 0.6 of coming up heads. Let X = 1 if the penny comes up heads, and let X = 0 if the penny comes up tails. Let Y = 1 if the nickel comes up heads, and let Y = 0 if the nickel comes up tails.

a.  Find the probability mass function of X.

b.  Find the probability mass function of Y.

c.   Is it reasonable to assume that X and Y are independent? Why?

d.   Find the joint probability mass function of X and Y.

2.  Two fair dice are rolled. Let X represent the number on the first die, and let Y represent the number on the second die. Find µXY.

3.  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.

a.  Find the joint probability mass function of X and Y.

b.  Find the marginal probability mass functions pX (x) and pY (y).

c.    Find µX and µY .

d.  Find µXY .

e.   Find Cov(X,Y).

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