An airline knows that there is a 95% chance that any passenger for a commuter flight that will hold 189 passengers will show up, and assumes passengers arrive independently of one another. The airline decides to sell n =199 tickets to reduce the number of empty seats, expecting 5% of the passengers not to show up. Let X be a random variable giving the number of people who show up for the flight, and let Y = X - 189 be a random variable for the difference between the number of passengers who show up and the number of seats on the plane.
a) Give expressions for E(X), V ar(X), E(Y ), V ar(Y ).
b) Give an expression for P(Y > 0). What do you think of their choice to sell 199 tickets?
c) They conduct a review of their policies and decide that they do not want the bad public relations associated with passengers with tickets not getting a seat. So they hire you as a consultant at an exorbitant fee to give them advice. After you probe, you learn that as long as they have enough seats for passengers with tickets 98% of the time, they will accept the risk. What is the largest value of n so that P(Y > 0) = 0.02?
d) Briefly explain why the Bernoulli process assumptions might not hold here.