During the 2010-2011 NBA season, LeBron James of the Miami Heat attempted 279 three-point baskets and made 92. He also attempted 1,485 two-point baskets, making 758 of these. Use these counts to determine probabilities for the following questions.
(a) Let a random variable X denote the result of a two-point attempt. X is either 0 or 2, depending on whether the basket is made. Find the expected value and variance of X.
(b) Let a second random variable Y denote the result of a three-point attempt. Find the expected value and variance of Y.
(c) In a game, LeBron attempts 5 three-point baskets and 20 two-point baskets. How many points do you expect him to score? (Hint: Use a collection of iid random variables, some distributed like X and some like Y.)
(d) During this season, LeBron averaged 22.7 points from two- and three-point baskets. In the game described in part (c), he made 3 three-pointers and 10 two-point baskets. Does this seem typical for his total? Be sure to identify any further assumptions that you make.