need part e f g only 7. [25 points] The passenger\'s perspective: Based on data
ID: 3354914 • Letter: N
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need part e f g only
7. [25 points] The passenger's perspective: Based on data collected by the U.S. Department of Transportation from 2010-20164, each year there were (approximately) 600,000,000 boarded passengers . 440,000 voluntary denied boarding passengers 60,000 involuntary denied boarding passengers While these aggregate statistics are useful, we want to better describe the experience of an individual airline passenger: "How likely am I to be bumped?" To do so, we model being bumped from a flight as a sequence of i.i.d. Bernoulli random variables, Bi, Ba,.. where B, = 1 if I am bumped and 0 otherwise. (We ignore voluntary denied boarding.) In parts (a)-(d) we first use the data above to parameterize the model (ie., find P(B -1) Then, in parts (e)-(g) analyze the model. (a) [1 point] How many passengers were involuntarily denied boarding for every 100,000 passengers who boarded? (b) [1 point] Assume that all flights have 200 seats and each flight has either 1 or 0 invol- untary denied boarding passenger with probability p. Find the value of p so that the expected number of involuntary denied boarding passengers over 500 flights matches your answer in (a). (c) [1 point] Suppose you are a passenger on one of these flights. Given that your flight is overbooked (i.e., there are 201 ticketed passengers for 200 seats), and assuming each passenger is equally likely to be bumped by the airline, what is the probability that you are bumped? (d) [I point] From (b) and (e), what is the probability that you are bumped from a flight? (e) [7 points) Let R be the number of fights you take before the first flight you are bumped from. What is the distribution of R? Calculate and interpret E l4R) (0) 18 points) Assume (and specify) some reasonable number of flights you would take per year. What is the probability you are bumped from a flight in the next 5 years? The next 10 years? (8) [6 points] State three model assumptions that you think are unrealistic and a brief explanation (one sentence) for each.Explanation / Answer
(e)
The distribution of R is a geometric distribution with the parameter p where p is the probability of bumped from the flight.
Probability of bumped from flight = Number of bumped passengers / Number of boarded passengers
= 60,000 / 600, 000, 000 = 0.0001
Therefore,
R ~ Geom(0.0001)
E[R] = 1/p = 1/0.0001 = 10,000
(f)
Let us assume, we take 10 number of flights each year. So, in next 5 years , we take 10*5= 50 flights.
In 50 flights, the probability to be bumped from flight = 1 - Probability to be never bumped in 50 flights
= 1 - (1 - 0.0001)50
= 1 - 0.999950
= 1 - 0.995
= 0.005
In next 10 years , we take 10*10 = 100 flights.
In 100 flights, the probability to be bumped from flight = 1 - Probability to be never bumped in 100 flights
= 1 - (1 - 0.0001)100
= 1 - 0.9999100
= 1 - 0.99
= 0.01
(g)
The three model assumptions of the Bernoulli process is,
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