1. A bus is scheduled to arrive at a bus stop at 8:30am. In reality, the time the bus arrives is uniformly distributed between 8:27am and 8:40am. Let X represent the number of minutes it takes, starting from 8:27am, for the bus to arrive to the bus stop. That is, X is uniformly distributed between 0 and 13 minutes.
(a) If you arrive to the bus stop at exactly 8:32am, what is the probability that you missed the bus?
(b) You arrive at 8:32am but another person at the bus stop says they have been there since 8:25am and the bus has not yet arrived. What is the probability you will wait less than 5 minutes for the bus?
2. Suppose that the time between mechanical failures occurring in an industrial plant follows an exponential distribution with a mean of 9.4 hours. Let X represent the time, in hours, between two mechanical failures. Compute the following probabilities.
(a) P(X > 12)
(b) P(5 < X < 6)
3. Analysis of data from 1915 to 2000 shows that the annual percentage change in the consumer price index (CPI; all items) in Canada can be described by a normal distribution with a mean of 3.36%, and a standard deviation of 5.1%. Let X represent the annual percentage change in CPI.
(a) Find the probability that the annual change in CPI will be between 1% and 4%.
(b) Find the probability that the annual change in CPI will exceed 12%.
(c) Fifteen percent of the time, the annual percent change in CPI is below what value? Is this result reasonable? Explain.
4. Divided Airlines makes use of the Airbus A380-800, the biggest passenger airplane in the world. The airplane provides seating for 525 passengers. Because the airline knows that, historically, 4% of all ticket holders will not show up for a flight, they purposely sell 540 tickets for each flight. Let X represent the total number of ticket holders who show up expecting a seat on a flight.
(a) Assuming that ticket holders show up for a flight independently of each other, what is the true probability distribution of X? State the values of all relevant parameters.
(b) What probability distribution can be used to approximate the probability distribution of X? Why? State the values of all relevant parameters.
(c) Approximate the probability that, for a given flight, that there will be a seat available for every ticket holder who shows up.