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).
