You drive to work 5 days a week for a full year (50 weeks), and with probability p = 0.02 you get a traffic ticket on any given day, independently of other days. Let X be 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) Any one 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 amount of money you pay in traffic tickets during the year.
(d) Suppose you don't know the probability p of getting a ticket, but you got 5 tickets during the year, and you estimate p by the sample mean p =5/250= 0.02. What is the range of possible values of p assuming that the difference between p and the sample mean ˆp is within 5 times the standard deviation of the sample mean?