Reference no: EM13672224
Problem 1.
The safe level of total coliform bacteria in well water is 5 or fewer bacteria per 100 ml. Then one rule for declaring a sample to be "safe" is if a 20 ml sample contains 0 or 1 bacteria. Assume coliform bacteria in well water are distributed according to a Poisson process at an average concentration of ? per 100 ml, and let X be a random variable giving the number of coliform bacteria in a 20 ml sample of well water.
a) If ? = 10 (twice the safe limit), what is the probability a 20 ml test tube will contain 0 or 1 bacteria (and so the sample will be declared "safe")?
b) Regulations require that three consecutive samples taken one week apart all have to be "safe" in order to declare the well "safe". If ? = 10 each week, what is the probability that all three 20 ml samples will show 0 or 1 bacteria?
c) A 200 ml container is used to collect a larger sample and this sample is transferred to ten 20 ml test tubes for analysis. If ? = 10 and the 200 ml sample contains 20 bacteria, derive the probability distribution for Y , the number of bacteria in the first test tube. Name the probability distribution for Y , and give E(Y ) and V ar(Y ).
d) The regulators feel that the current regulations requiring 3 safe samples taken a week apart are not protective enough if ? = 10. They are thinking of changing the rules to either defining a safe sample as one with 0 bacteria and requiring a safe sample each week for three weeks, or keeping the definition of a safe sample as it is but requiring a safe sample each week for n weeks. Find the smallest value of n so that, if ? = 10, P(declare a sample safe under Rule 2) < P(declare a sample safe under Rule 1)
Problem 2.
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.