In basketball, there are four quarters in a game. Suppose that when UCSB plays another team that has roughly the same record of play, the number of points scored in a quarter by the home team minus the number scored by the visiting team is approximately a normal random variable with mean 1.5 and variance 6. In addition, suppose that the point differentials for the four quarters are independent. Hint: Let Xi = Hi − Vi where Hi is the number of points scored by the home team in quarter i, Vi is the number of points scored by the visiting team in quarter i, and Xi is the number of points scored by the home team minus the number scored by the visiting team in quarter i.
(a) What is the probability that the home team wins?
(b) What is the conditional probability that the home teams wins, given that it is behind by 5 points at halftime?
(c) What is the conditional probability that the home team wins, given that it is ahead by 5 points at the end of the first quarter?
10. Let X be a continuous random variable with density function