Question 1. Air America has a policy of routinely overbooking flights. The random variable X represents the number of passengers who cannot be boarded because there are more passengers than seats. The probability mass function of X is as follows:
(i) What is the probability that the number of passengers who cannot be boarded is more than 2 in a randomly selected flight?
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x 0
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1 2 3 4
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5
|
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P (X = x) 0.55
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0.25 0.10 0.05 0.03
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0.02
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(ii) Write down the cumulative distribution function of X.
(iii) Calculate the mean and standard deviation of X.
(iv) In a random sample of six flights, what is the probability that all the passengers can be boarded in at least half of flights in the sample?
(v) The airline also has another policy of compensation for any passenger affected by overbooking flights. The airline will pay $80 per passenger who cannot be boarded due to overbooking in any flight. Let Y be the total amount of compensations paid to all passengers who cannot be boarded in one particular flight. Write down the probability mass function of Y .
Question 2. Number of motor vehicle death events per month in Dutchess County, New York, is modeled by a Poisson distribution with average of 1.8 per month.
(i) What is the probability that there are more than 3 motor vehicle death events in Dutchess County in a given month?
(ii) What is the probability that in a particular year, there are between 24 to 27 motor vehicle death events, inclusive, in Dutchess County?
(iii) What is the probability that in a given year, there are exactly two months with more than 3 motor vehicle death events per month (it implies that each of the other ten months has less than or equal to 3 motor vehicle death events) ?
Question 3. A zoologist has ten male and seven female guinea pigs. She randomly selects five guinea pigs for an experiment.
(i) What is the probability that more female guinea pigs are selected in the experiment?
(ii) Given that more female guinea pigs are selected in the experiment, what is the probability that exactly four female guinea pigs are selected in the experiment?
Question 4. You are trying to set up a portfolio that consists of Fund X and Fund Y . The following information about the annual return (per $1000) of each of these investments under different economic conditions is available along with the probability that each of these economic conditions will occur in next year.
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Probability
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State of Economy Fund X ($) Fund Y ($)
|
|
0.10
|
Recession
|
-160
|
350
|
|
0.15
|
Stagnation
|
-40
|
100
|
|
0.35
|
Slow growth
|
80
|
40
|
|
0.30
|
Moderate growth
|
150
|
-30
|
|
0.10
|
High growth
|
200
|
-100
|
(i) Compute the expected return for the Fund X and for the Fund Y .
(ii) Compute the standard deviation for the Fund X and for the Fund Y .
(iii) Compute the covariance of the Fund X and the Fund Y .
(iv) Suppose that you want to create a portfolio that consists of 25% Fund X and 75% Fund Y . Compute the portfolio expected return and portfolio risk.
(v) Find the probability that your portfolio return is more than $45 per $1000 invest- ment in the next year.