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, independent 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 of the tickets is $10 or $20 or $50 with respective probabilities 0.5, 0.3, and 0.2, and independent of other tickets. Find the mean and the 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
What is the range of possible values of p assuming that the difference between p and the sample mean p is within 5 times the standard deviation of the sample mean?