Statistics, winter semester 20–21
M. Raux
Exercise Set I
Probability fundamentals
1. (Airline example from Wooldridge, Appendix B). A flight has 100 available seats, and the airline sells
more tickets than available seats. Given that the probability that each person shows up is 𝜃 = .85, if
the airline sells 110 tickets, what is the probability that more than 100 travellers show up? Write the
formula, you do not need to compute the exact value of the probability.
2. (Wooldridge, exercise B2) Much is made of the fact that certain mutual funds outperform the market
year after year (that is, the return from holding shares in the mutual fund is higher than the return
from holding a portfolio such as the S&P 500). For concreteness, consider a 10-year period and let the
population be the 4170 mutual funds reported in The Wall Street Journal on January 1, 1995. By saying
that performance relative to the market is random, we mean that each fund has a 50–50 chance of
outperforming the market in any year and that performance is independent from year to year.
(i) If performance relative to the market is truly random, what is the probability that any particular
fund outperforms the market in all 10 years?
(ii) Find the probability that at least one fund out of 4170 funds outperforms the market in all 10
years. What do you make of your answer?
(iii) Find the probability that at least five funds outperform the market in all 10 years (Write the
formula, you do not need to compute the exact value of the probability).
3. An airline opens the sale of tickets for a flight. Before starting to sell the tickets, the airline does not
know the proportion of passengers that will buy a business class ticket, so this is also a random variable.
The following tables gives the probabilities that a passenger actually shows up after having bought a
flight ticket, and whether he is an economy or business traveler. The definitions of the variables are
𝑋 =
0
1
if the passenger shows up,
if the passenger does not shows up,
𝑌 =
0
1
buys economy class,
buys business class.
The following two tables give the probability density functions 𝑃𝑌 |𝑋 (𝑦|𝑥) and 𝑃𝑋𝑌 (𝑥, 𝑦) :
𝑃𝑌 |𝑋 (𝑦|𝑥)
Y
(i)
0
1
0
0.3
0.7
X
𝑃𝑋𝑌 (𝑥, 𝑦)
1
0.2
0.8
0
0.03
0.07
0
1
Y
Compute 𝑃𝑋 (𝑥) .
(ii) Can you compute 𝑃𝑋 |𝑌 (𝑥 |𝑦) ?
(iii) Let 𝑃𝑋 (𝑥) be the ones computed in point. (i) Let 𝑃𝑌 |𝑋 (𝑦|𝑥) now be:
𝑃𝑌 |𝑋 (𝑦|𝑥)
Y
0
1
0
0.2
0.8
X
1
0.2
0.8
X
1
0.18
0.72
Can you compute 𝑃𝑋 |𝑌 (𝑥 |𝑦) ?
4. (Wooldridge, exercise B4) For a randomly selected county in the United States, let 𝑋 represent
the proportion of adults over age 65 who are employed, or the elderly employment rate. Then, 𝑋 is
restricted to a value between zero and one. Suppose that the cumulative distribution function for 𝑋 is
given by 𝐹 (𝑥) = 3𝑥 2 − 2𝑥 3 , for 0 ≤ 𝑥 ≤ 1. Find the probability that the elderly employment rate is at
least 0.6 ( 60%) .
5. (Wooldridge, exercise B6) Let 𝑋 denote the prison sentence, in years, for people convicted of auto
theft in a particular state in the United States. Suppose that the pdf of 𝑋 is given by
1
9
𝑓𝑋 (𝑥) = 𝑥 2,
0 < 𝑥 < 3.
What is the expected prison sentence.
6. You observe a stock price for 3 days in a row. Each day the stock price can go up by an amount 𝑢 or
down by an amount 𝑑 .
(i)
Write the sample space 𝑆 for this experiment. How many elements does it contain?
(ii) Assume that 𝑢 = −𝑑 . Does the sample space change? How many elements does it have?
(iii) Assume that the price of the stock at day 0 is 10, and that 𝑢 = 1 and 𝑑 = −2. Let 𝑍 be the value
of the stock at the end of the 3 days. What is the sample space given by the outcomes of 𝑍 ?
(iv) I bought the stock at day 0 and I am selling it on day 3 only if I make a profit, that is if the price
is strictly larger than 10. If each day the probabilities of the stock moving up or down are equal,
what is the probability that I sell the stock on day 3?
7. (Wooldridge, exercise B7) If a basketball player is a 74% free throw shooter, then, on average, how
many free throws will he or she make in a game with eight free throw attempts?
8. (Based on Example 1.5.4. De Groot Schervish) Demands for Utilities. A contractor is building an
office complex and needs to plan for water and electricity demand (sizes of pipes, conduit, and wires).
After consulting with prospective tenants and examining historical data, the contractor decides that the
demand for electricity will range somewhere between 1 million and 150 million kilowatt-hours per day
and water demand will be between 4 and 200 (in thousands of gallons per day). All combinations of
electrical and water demand are considered possible. Let 𝑋 be the demand of water and 𝑌 the demand
of electricity. The contractor is interested in these two events:
high water demand, 𝐸 1 : {(𝑥, 𝑦) | 𝑥 ≥ 100}, and
high electricity demand, 𝐸 2 : {(𝑥, 𝑦) | 𝑦 ≥ 115}.
Assume that the relative probabilities of the events are given by the ratios of the surfaces of the 𝑋𝑌
plane defined by the events.
(i)
What is the sample space 𝑆 ? What is 𝑃 (𝑆) ?
(ii) What is the probability of 𝐸 1 ?
(iii) What is the probability of 𝐸 2 ?
(iv) What is the probability that the contractor has to face both a high demand in electricity and a
high demand in water?
(v) What is the probability that a the contractor will have to assure a high level of water supply
conditional to tenants asking for a high electricity supply, that is 𝑃𝑋 |𝑌 (𝑋 = 1 |𝑌 = 1) ?
(vi) Are the variables 𝑋 and 𝑌 independent? Explain.
9. Consider the picture below. Variables and interpretations are the same as in exercise 8. The hatched
rectangle denotes commercial tenancies. Define the variable 𝑍 , where 𝑍 = 0 if the tenant is residential
and 𝑍 = 1 if he is commercial. If we restrain our attention to residential tenancies, that is, if we condition
the distributions on 𝑍 = 0, are 𝑋 and 𝑌 conditionally independent? Formally, this is written as
𝑋 and 𝑌 are independent given 𝑍 if 𝑃𝑋𝑌 |𝑍 (𝑥, 𝑦|𝑧) = 𝑃𝑋 |𝑍 (𝑥 |𝑧)𝑃𝑌 |𝑍 (𝑦|𝑧) .
Electricity (Y)
150
115
1
4
100
200
Water (X)
10. Suppose that at a large university, college grade point average, 𝐺𝑃𝐴, and SAT (a college admission
test) score, 𝑆𝐴𝑇 , are related by the conditional expectation 𝐸 (𝐺𝑃𝐴|𝑆𝐴𝑇 ) = .70 + 0.002 𝑆𝐴𝑇 .
(i)
Find the expected 𝐺𝑃𝐴 when 𝑆𝐴𝑇 = 800. Find 𝐸 (𝐺𝑃𝐴|𝑆𝐴𝑇 = 1400) . Comment on the
difference.
(ii) If the average 𝑆𝐴𝑇 in the university is 1100, what is the average GPA?
(iii) If a student’s 𝑆𝐴𝑇 score is 1100, does this mean he or she will have the 𝐺𝑃𝐴 found in part (ii)?
Explain.
11. (Advanced) Show that if 𝐷 is a Bernoulli variable, and 𝑋 a generic variable defined on the same
sample space as 𝐷 , then:
𝐸 (𝑋 |𝐷 = 1) =
𝐸 (𝑋 𝐷)
𝐸 (𝑋 𝐷)
=
.
𝐸 (𝐷)
𝑃 (𝐷 = 1)
(Hint. 𝑋 and 𝐷 are defined on the same sample space 𝑆 , so 𝐷 is a partition of 𝑆 . Split the integral of the
definition of 𝐸 (𝑋 𝐷) according to the regions given by this partition, and the result follows).
12. Let 𝑋 and 𝑌 be random variables such that
𝑋
3
1 −1
∼N
,
.
𝑌
1
−1 4
(i)
Define what it means for 𝑋 and 𝑌 to be jointly normal.
(ii) Find the joint distribution of 𝑋 + 2𝑌 and 2𝑋 − 𝑌 .
(iii) Calculate 𝑃 (𝑋 + 21 𝑌 ≤ 0) .
(iv) Find 𝐸 [𝑌 |𝑋 ] and Var (𝑌 |𝑋 ) .
(v) Compare Var (𝑌 |𝑋 = 1) with Var (𝑌 |𝑋 = −1/2) . Is there any difference? Explain why.
13. 𝑈 and 𝑉 are standard normal random variables, and Corr (𝑋, 𝑌 ) = 0.5.
(i)
2𝑈
Write the variance-covariance matrix of
3𝑉
(ii) Choose 𝑎 1, 𝑎 2 such that 𝑊 = 𝑎 1𝑈 + 𝑎 2𝑉 and 𝑈 are independent.
(iii) Choose 𝑏 1, 𝑏 2 such that 𝑍 = 𝑏 1𝑊 2 + 𝑏 2𝑈 2 follows a 𝜒 22 distribution.