You drive to work 5 days a week for a full year (50 weeks), and on any given day, you get a traffic ticket with probabilityp= 0.02, independently of other days. LetXbe the total number of tickets you get in the year.

(a) What is the probability that the number of tickets you get is exactly equal to the expected value of X?

(b) Calculate approximately the probability in (a), using a Poisson approximation.

(c) The fine for a ticket is $10, or $20, or $50, with respective probabilities 0.5, 0.3, and 0.2, and independently of other tickets. Find the mean and variance of the total amount you pay, which we denote byY, for traffic tickets during the year?

(d) Suppose you do not know the probabilitypof getting a ticket, but you got 5 tickets

during the year, and you estimatepby the sample meanP= 5/250. What is the range of

possible values ofpassuming that the difference betweenpand the sample meanPis within 5 times the standard deviation of the sample mean? (see Example 2.21 in the textbook for information on the use and properties of the sample mean, in particular, its variance.)