ProblemSet Random Variables

Problem Set
(Courtsey: Aczel & Sounderpandian)
Discrete Random Variables

1. The number of telephone calls arriving at an exchange during any given minute between noon
and 1:00 P.M. on a weekday is a random variable with the following probability distribution.
x
0
1
2
3
4
5
P(x)
0.3
0.2
0.2
0.1
0.1
0.1
x
0
1
2
3
4
5
6
P(x)
0.01
0.09
0.30
0.20
0.20
0.10
0.10
a. Verify that P(x) is a probability distribution.

b. Find the cumulative distribution function of the random variable.
c. Use the cumulative distribution function to find the probability that between 12:34 and
12:35 P.M. more than two calls will arrive at the exchange.
2. According to an article in Travel and Leisure, every person in a small study of sleep during
vacation was found to sleep longer than average during the first vacation night. 1 Suppose that the
number of additional hours slept in the first night of a vacation, over the persons average number
slept per night, is given by the following probability distribution:

b. Find the cumulative distribution function.
c. Find the probability that at most four additional hours are slept.
d. Find the probability that at least two additional hours are slept per night.
3. The percentage of people (to the nearest 10) responding to an advertisement is a random
variable with the following probability distribution:
x(%)
0
10
20
30
40
50
P(x)
0.10
0.20
0.35
0.20
0.10
0.05
a. Show that P(x) is a probability distribution.

b. Find the cumulative distribution function.
c. Find the probability that more than 20% will respond to the ad.
4. An automobile dealership records the number of cars sold each day. The data are used in
calculating the following probability distribution of daily sales:
x
0
1
2
3
4
5
P(x)
0.1
0.1
0.2
0.2
0.3
0.1
a. Find the probability that the number of cars sold tomorrow will be between two and four
(both inclusive).
b. Find the cumulative distribution function of the number of cars sold per day.
c. Show that P(x) is a probability distribution.
5. Consider the roll of a pair of dice, and let X denote the sum of the two numbers appearing on
the dice. Find the probability distribution of X, and find the cumulative distribution function. What is
the most likely sum?
6. The number of intercity shipment orders arriving daily at a transportation company is a random
variable X with the following probability distribution:
x
0
1
2
3
4
5
P(x)
0.1
0.2
0.4
0.1
0.1
0.1

b. Find the cumulative probability function of X.
c. Use the cumulative probability function computed in (b) to find the probability that
anywhere from one to four shipment orders will arrive on a given day.
d. When more than three orders arrive on a given day, the company incurs additional costs
due to the need to hire extra drivers and loaders. What is the probability that extra costs
will be incurred on a given day?
e. Assuming that the numbers of orders arriving on different days are independent of each
other, what is the probability that no orders will be received over a period of five working
days?
f. Again assuming independence of orders on different days, what is the probability that
extra costs will be incurred two days in a row?
7. An article in The New York Times reports that several hedge fund managers now make more
than a billion dollars a year.2 Suppose that the annual income of a hedge fund manager in the top
tier, in millions of dollars a year, is given by the following probability distribution:
x ($ millions)
$1,700
1,500
1,200
1,000
800
600
400
P(x)
0.2
0.2
0.3
0.1
0.1
0.05
0.05
a. Find the probability that the annual income of a hedge fund manager will be between
$400 million and $1 billion (both inclusive).
b. Find the cumulative distribution function of X.
c. Use F(x) computed in (b) to evaluate the probability that the annual income of a hedge
fund manager will be less than or equal to $1 billion.
d. Find the probability that the annual income of a hedge fund manager will be greater than
$600 million and less than or equal to $1.5 billion.
8. The number of defects in a machine-made product is a random variable X with the following
probability distribution:
x
0
1
2
3
4
P(x)
0.1
0.2
0.3
0.3
0.1
a. Show that P(x) is a probability distribution.

b. Find the probability P(
c. Find the probability P(
1< X 3
1< X 3
).
).
d. Find F(x).
9. The daily exchange rate of one dollar in euros during the first three months of 2007 can be
inferred to have the following distribution.
x
0.73
0.74
0.75
0.76
0.77
0.78
P(x)
0.05
0.10
0.25
0.40
0.15
0.05
a. Show that P (x) is a probability distribution.

b. What is the probability that the exchange rate on a given day during this period will be
at least 0.75?
c. What is the probability that the exchange rate on a given day during this period will be
less than 0.77?
d. If daily exchange rates are independent of one another, what is the probability that for
two days in a row the exchange rate will be above 0.75?
10. According to Chebyshevs theorem, what is the minimum probability that a random variable will
be within 4 standard deviations of its mean?
11. At least eight-ninths of a population lies within how many standard deviations of the population
mean? Why?
12. Management of an airline knows that 0.5% of the airlines passengers lose their luggage on
domestic flights. Management also knows that the average value claimed for a lost piece of
luggage on domestic flights is $600. The company is considering increasing fares by an appropriate
amount to cover expected compensation to passengers who lose their luggage. By how much
should the airline increase fares? Why? Explain, using the ideas of a random variable and its
expectation.
Binomial & Poisson Distribution

1.
2.
3.
4.
5.
6.
A salesperson finds that, in the long run, two out of three sales calls are successful. Twelve
sales calls are to be made; let X be the number of concluded sales. Is X a binomial random
variable? Explain.
A large shipment of computer chips is known to contain 10% defective chips. If 100 chips
are randomly selected, what is the expected number of defective ones? What is the
standard deviation of the number of defective chips? Use Chebyshevs theorem to give
bounds such that there is at least a 0.75 chance that the number of defective chips will be
within the two bounds.
A new treatment for baldness is known to be effective in 70% of the cases treated. Four
bald members of the same family are treated; let X be the number of successfully treated
members of the family. Is X a binomial random variable? Explain.
What are Bernoulli trials? What is the relationship between Bernoulli trials and the binomial
random variable?
A salesperson goes door-to-door in a residential area to demonstrate the use of a new
household appliance to potential customers. At the end of a demonstration, the probability
that the potential customer would place an order for the product is a constant 0.2107. To
perform satisfactorily on the job, the salesperson needs at least four orders. Assume that
each demonstration is a Bernoulli trial.
a. If the salesperson makes 15 demonstrations, what is the probability that there would be
exactly 4 orders?
b. If the salesperson makes 16 demonstrations, what is the probability that there would be
at most 4 orders?
c. If the salesperson makes 17 demonstrations, what is the probability that there would be
at least 4 orders?
d. If the salesperson makes 18 demonstrations, what is the probability that there would be
anywhere from 4 to 8 (both inclusive) orders?
e. If the salesperson wants to be at least 90% confident of getting at least 4 orders, at least
how many demonstrations should she make?
f. The salesperson has time to make only 22 demonstrations, and she still wants to be at
least 90% confident of getting at least 4 orders. She intends to gain this confidence by
improving the quality of her demonstration and thereby improving the chances of getting
an order at the end of a demonstration.
At least to what value should this probability be increased in order to gain the desired
confidence? Your answer should be accurate to four decimal places.
A commercial jet aircraft has four engines. For an aircraft in flight to land safely, at least
two engines should be in working condition. Each engine has an independent reliability of p
_ 92%.
a. What is the probability that an aircraft in flight can land safely?
b. If the probability of landing safely must be at least 99.5%, what is the minimum
value for p? Repeat the question for probability of landing safely to be 99.9%.
7.
c. If the reliability cannot be improved beyond 92% but the number of engines in a
plane can be increased, what is the minimum number of engines that would
achieve at least 99.5% probability of landing safely? Repeat for 99.9% probability.
d. One would certainly desire 99.9% probability of landing safely. Looking at the
answers to (b) and (c ), what would you say is a better approach to safety,
increasing the number of engines or increasing the reliability of each engine?
A real estate agent has four houses to sell before the end of the month by contacting
prospective customers one by one. Each customer has an independent 0.24 probability of
buying a house on being contacted by the agent.
a. If the agent has enough time to contact only 15 customers, how confident can
she be of selling all four houses within the available time?
b. If the agent wants to be at least 70% confident of selling all the houses within
the available time, at least how many customers should she contact? (If necessary,
extend the template downward to more rows.)
c. What minimum value of p will yield 70% confidence of selling all four houses by
contacting at most 15 customers?
d. To answer (c) above more thoroughly, tabulate the confidence for p values
ranging from 0.2 to 0.6 in steps of 0.05.
8. The number of rescue calls received by a rescue squad in a city follows a

Poisson distribution with m=2.83 per day. The squad can handle at most
four calls a day.
a. What is the probability that the squad will be able to handle all
the calls on a particular day?
b. The squad wants to have at least 95% confidence of being able to
handle all the calls received in a day. At least how many calls a day
should the squad be prepared for?
c. Assuming that the squad can handle at most four calls a day,
what is the largest value of m that would yield 95% confidence that
the squad can handle all calls?
9. A mainframe computer in a university crashes on the average 0.71 time in
a semester.
a. What is the probability that it will crash at least two times in a
given semester?
b. What is the probability that it will not crash at all in a given
semester?
c. The MIS administrator wants to increase the probability of no
crash at all in a semester to at least 90%. What is the largest that
will achieve this goal?

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Problem Set

(Courtsey: Aczel & Sounderpandian)

Discrete Random Variables

a. Verify that P(x) is a probability distribution.

a. Verify that P(x) is a probability distribution.

a. Show that P(x) is a probability distribution.

a. Verify that P(x) is a probability distribution.

a. Show that P(x) is a probability distribution.

a. Show that P (x) is a probability distribution.

Binomial & Poisson Distribution

8. The number of rescue calls received by a rescue squad in a city follows a

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