During the 2011-2012 NFL regular season, kicker Lawrence Tynes of the Super Bowl winning New York Giants attempted 24 field goals, making 19. Of the kicks, he made 15 out of 16 that were for less than 40 yards, and he made 4 out of 8 longer kicks. Use these counts to form probabilities.
(a) Let the random variable X denote the points scored when Tynes attempts a field goal of less than 40 yards. If he makes it, the Giants score 3 points. If he misses, they get none. Find the mean and variance of X.
(b) Similarly, let Y denote the points scored when Tynes attempts a field goal of more than 40 yards. Find the mean and variance of Y.
(c) During a game, Tynes attempts three kicks of less than 40 yards and two of more than 40 yards. How many points do you expect him to contribute to the total for the Giants?
(d) During the season, Tynes scored 57 points on field goals, about 3.6 per game. If Tynes scores 15 points on kicks in the game described in (c), would that be an unusually large contribution to the total score?