1. Airlines often oversell tickets. Suppose that a plane with 100 seats, 104 passengers have tickets. Let X be the number of ticketed passengers who actually show up for the flight. Assume that the probability mass function of X is given in the following table X 96 97 98 99 100 101 102 103 104 17 .06.05 p(x) .15 12 14 25 .04 .02 (a) Draw the probability histogram for the distribution of X. (b) What are the probabilities P(X ≤ 100) and P(98 < X < 102)? (c) If every passenger who does not get a seat needs to be compensated for $300, whereas every ticket was sold for $200. Let Z be the profit or loss of the airline due to overselling of 4 tickets, namely Z = 4*200 - (X-100)*300 if X> 100 and Z = 800 otherwise. What are the expected profit (EZ) and its associated standard deviation (SD(Z))? (Hint: As we have learnt in the class, a discrete random variable is completely specified by the value it takes and the associated probability vector. Try to find out what are the possible values of Z and the corresponding probabilities.)
1. Airlines often oversell tickets. Suppose that a plane with 100 seats, 104 passengers have tickets. Let X be the number of ticketed passengers who actually show up for the flight. Assume that the probability mass function of X is given in the following table X 96 97 98 99 100 101 102 103 104 17 .06.05 p(x) .15 12 14 25 .04 .02 (a) Draw the probability histogram for the distribution of X. (b) What are the probabilities P(X ≤ 100) and P(98 < X < 102)? (c) If every passenger who does not get a seat needs to be compensated for $300, whereas every ticket was sold for $200. Let Z be the profit or loss of the airline due to overselling of 4 tickets, namely Z = 4*200 - (X-100)*300 if X> 100 and Z = 800 otherwise. What are the expected profit (EZ) and its associated standard deviation (SD(Z))? (Hint: As we have learnt in the class, a discrete random variable is completely specified by the value it takes and the associated probability vector. Try to find out what are the possible values of Z and the corresponding probabilities.)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![1. Airlines often oversell tickets. Suppose that a plane with 100 seats, 104 passengers have tickets. Let \( X \) be the number of ticketed passengers who actually show up for the flight. Assume that the probability mass function of \( X \) is given in the following table:
\[
\begin{array}{c|cccccccccccc}
x & 96 & 97 & 98 & 99 & 100 & 101 & 102 & 103 & 104 \\
\hline
p(x) & .15 & .12 & .14 & .25 & .17 & .06 & .05 & .04 & .02 \\
\end{array}
\]
(a) Draw the probability histogram for the distribution of \( X \).
(b) What are the probabilities \( P(X \leq 100) \) and \( P(98 < X \leq 102) \)?
(c) If every passenger who does not get a seat needs to be compensated for $300, whereas every ticket was sold for $200. Let \( Z \) be the profit or loss of the airline due to overselling of 4 tickets, namely \( Z = 4 \times 200 - (X - 100) \times 300 \) if \( X > 100 \) and \( Z = 800 \) otherwise. What are the expected profit (\( E(Z) \)) and its associated standard deviation (\( SD(Z) \))? (Hint: As we have learnt in the class, a discrete random variable is completely specified by the value it takes and the associated probability vector. Try to find out what are the possible values of \( Z \) and the corresponding probabilities.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77ccc228-61ae-45bc-bff3-36e89ce5abe9%2F68746d85-5d11-4ac1-8d68-a139221deb8f%2Fp5vltcd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Airlines often oversell tickets. Suppose that a plane with 100 seats, 104 passengers have tickets. Let \( X \) be the number of ticketed passengers who actually show up for the flight. Assume that the probability mass function of \( X \) is given in the following table:
\[
\begin{array}{c|cccccccccccc}
x & 96 & 97 & 98 & 99 & 100 & 101 & 102 & 103 & 104 \\
\hline
p(x) & .15 & .12 & .14 & .25 & .17 & .06 & .05 & .04 & .02 \\
\end{array}
\]
(a) Draw the probability histogram for the distribution of \( X \).
(b) What are the probabilities \( P(X \leq 100) \) and \( P(98 < X \leq 102) \)?
(c) If every passenger who does not get a seat needs to be compensated for $300, whereas every ticket was sold for $200. Let \( Z \) be the profit or loss of the airline due to overselling of 4 tickets, namely \( Z = 4 \times 200 - (X - 100) \times 300 \) if \( X > 100 \) and \( Z = 800 \) otherwise. What are the expected profit (\( E(Z) \)) and its associated standard deviation (\( SD(Z) \))? (Hint: As we have learnt in the class, a discrete random variable is completely specified by the value it takes and the associated probability vector. Try to find out what are the possible values of \( Z \) and the corresponding probabilities.)
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