PS4

Assignment 4 Economics 31 Fall 1999

More Expected Values and Probability Distributions

Reading Assignment: Mirer Chpt. 9, Beals Chpt. 3,4,5.

1. The Beloit Corporation, which sells paper manufacturing equipment, has
only one salesman, whose income depends on
the number of machines he sells per day. Specifically, he receives no
commission on the first machine sold per day, a $20 commission on the
second machine sold in a day, a $30 commission on the third, and a $40
commission on the fourth. (Thus, if he sells 3 machines in a given day, his
commissions for this day total $50.) His income consists entirely of these
commissions. It is equally likely that he will sell 0,1, 2,3, or 4 paper
manufacturing machines in a day.

a)What values can his income assume? Construct a table showing its
probability distribution and plot the probability distribution of the
salesman's income on a particular day in a line chart.

(b) What is the probability that his income in a particular day will exceed
$20? $30? $40?

(c) Suppose that the number of machines sold one day is independent of the
number sold the next day. Construct a table showing the probability
distribution of the total number of machines sold in a two-day period. What
is the probability that this number will exceed 2? 4? 6? 8?

(d) What is the probability the salesman will earn $100 over two days?

(e) Under the circumstances described above, what is the expected value of
the income of the Beloit Corporation's salesman in a particular day?

2. A salesman owes money to Crazy TV Lenny at American TV in Madison,
Wisconsin for a recently purchased stereo. Crazy TV Lenny offers our
salesman the option of either paying the 500 dollars he owes now or
flipping a coin twice for an immediate fee of 25 dollars. In the event that
our salesman elected to take the gamble and received two heads, his 500
dollar debt would be cut in half

(a) Is this a fair gamble?

(b) What is the expected value of the salesman's winnings (or Losses) in
this game?

3. There is a .23 probability that a United Airlines flight flying from
Philadelphia to Chicago O'Hare between the days Dec 19th and 21st will be
greater than 15 minutes late, with a .77 probability of being on time.
There are seven United flights per day.
(http://www.bts.gov/cgi-bin/oai/ontime_js.pl)

(a) What is the expected number of late flights on Dec 19th? Is this a
value that the random variable in question can assume?

(b) What is the expected number of late flights between Dec. 19th and 21st?

4. The probability of being late on one flight is independant of the
probability of being late on another flight.

a) Assuming that United flights arrive exactly 15 minutes late when they
are late, and that you place the numerical value your time at approximately
$.30 per minute, and that a flight from Philly to O'Hare between Dec 19th
and 21st costs exactly$250, using the probabilities above calculate the
implicit expected cost of your flight.

b) Fortunately, both American and USAir also offer seven flights per day
from Philly to O'Hare. American flights have a probability of .11 of being
late, whereas USAir flights arrive late to O'Hare 17.54 percent of the
time. Unfortunately, however, both American and USAir flights cost $300.
Which flight should you take?

c) Flights from Philly to Midway, another Chicago airport, are generally
cheaper. Given that Midway is a smaller, dingier airport, located farther
from your home, with poor public transportation facilities, how much
cheaper would the fare have to be, to induce you to fly to Midway
(recognizing that traffic jams are much more frequent near Midway than
O'Hare)? Do you think that as an adult, this figure might adjust? Why?

5. A raffle box contains 100 certificates worth 0 each, 10 certificates
worth $20 each, 5 certificates worth $50 each, and 1 certificate worth
$100.

(a) If one certificate is chosen at random, what is the expected value of its worth?

(b) What is the standard deviation of its worth?

(c) In this case, is it possible for the random variable (the worth of the
chosen certificate) to equal the expected value of the random variable?

(d) If you want to take the action that maximizes expected gain, should you
pay $2 for the opportunity to pick a certificate at random from this bag?

6. 10.2 percent represents the mean return of stock portfolios managed by
James. The standard deviation of these portfolios is 0.9 percent. Manager
James says that the probability of a stock portfolio's return exceeding
12.0% or being less than 8.4 percent is less than 0.25? Is he correct?
Does this spread seem unusual to you?

7. Sam Adams is considering initiating a program, A, for a large number of
unemployed workers. If program A is successful, taxpayers would earn 5,000
dollars in income taxes. If, however, program A is unsuccessful, the
government would have to pay out an estimated 10,000 dollars total in
unemployment compensation. Sam, a sophisticated economist, estimates
program A has a 55 percent success rate. Should Sam advise the government
to initiate program A?

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