Data Science 401 ASU
Exercises Hahn
1. Simulate the Monty Hall game from the first set of slides in R using the sample()
function. “Play” the game 10,000 times never switching and record how often you win.
Then “play” the game another 10,000 times always switching and record how often
you win. Which strategy is better?
2. Implement the code from page 13 on the week 2 slides for computing the expected
utility in the milk ordering problem. To make sure you understand the code, try
writing a version that uses for loops rather than the sapply command. Compute the
expected utility (of the optimal order) if you are provided with a forecast of if it will
rain or not and compare it to the expected utility if you do not have the ability to
know the weather in advance. Suppose that in the past 100 days, 20 have been rainy
and that on those days the number of gallons needed was as follows: there was one day
when 6 gallons were needed, one day when 9 gallons were needed and one day when 10
gallons were needed. There were two days when 7 gallons were needed and 15 where
8 gallons were needed. (Use this data as if it provides exact probabilities, rather than
estimates.)
3. Answer the following statements TRUE or FALSE, providing a succinct explanation
of your reasoning.
(a) You roll two fair three-sided dice. The probability the two dice show the same
number is 1/3.
(b) If events A and B are independent and P(A) > 0 and P(B) > 0, then P(A and B) >
0.
(c) If two events A and B are independent and P(A) > 0 and P(B) > 0, then A and
B cannot be mutually exclusive.
(d) If P(A and B) ≥ 0.4 then P (A) ≤ 0.40.
4. Prove the following properties of expectation directly from the definition. You may
assume a finite, discrete sample space for simplicity.
(a) E(aX) = aE(X)
(b) E(a + X) = a + E(X)
(c) E(X + Y) = E(X) + E(Y)
5. Come up with a counterexample to show that E(XY ) 6= E(X)E(Y ).
6. Show that, if X ⊥
⊥ Y , then E(XY ) = E(X)E(X).
7. Prove the following properties of variance in the finite, discrete sample space case.
1
, (a) V(aX) = a2 V(X)
(b) V(a + X) = V(X)
(c) V(aX + bY ) = a2 V(X) + b2 V(Y ) + 2abCov(X, Y )
8. Derive the identity: V (X) = E(X 2 ) − E(X)2 .
p
9. Consider the random variable Z = (X − E(X))/ (V(X)). Show that E(Z) = 0 and
V(Z) = 1.
10. Show that if two random variables are independent, then the correlation between them
is zero. Then, construct a counter-example to show that two dependent random vari-
ables can have zero correlation.
11. Show that E(X) = E(E(X | Y )) where the inner expectation is with respect to the
random variable X for a fixed value of Y and the outer expectation is over the random
variable Y . That is, show that
( )
X X
E(X) = p(y) x p(x | y) .
y x
This is called the iterated expectation formula.
12. An oil company wants to drill in a new location. A preliminary geological study
suggests that there is a 20% chance of finding a small amount of oil, a 50% chance
of a moderate amount and a 30% chance of a large amount of oil. The company has
a choice of either a standard drill that simply burrows deep into the earth or a more
sophisticated drill that is capable of horizontal drilling and can therefore extract more
but is far more expensive. The following table provides the payoff table in millions of
dollars under different states of the world and drilling conditions
small medium large
standard $20M $30M $50M
horizontal -$20M $40M $90M
(a) Find the mean payoffs of the two different drilling strategies.
(b) Find the variance in payoffs of each strategy.
(c) Which strategy would you advocate for and why?
(d) How much are you willing to pay for a geological evaluation that would determine
with certainty the quantity of oil at the site prior to drilling?
13. In a sample of 100,000 emails you find that 550 are spam. Your next email contains the
word “bigger”. From historical experience, you know half of all spam email contains
the word “bigger” and only 2% of non-spam emails contain it. Find the probability
that this new email is spam.
2
Exercises Hahn
1. Simulate the Monty Hall game from the first set of slides in R using the sample()
function. “Play” the game 10,000 times never switching and record how often you win.
Then “play” the game another 10,000 times always switching and record how often
you win. Which strategy is better?
2. Implement the code from page 13 on the week 2 slides for computing the expected
utility in the milk ordering problem. To make sure you understand the code, try
writing a version that uses for loops rather than the sapply command. Compute the
expected utility (of the optimal order) if you are provided with a forecast of if it will
rain or not and compare it to the expected utility if you do not have the ability to
know the weather in advance. Suppose that in the past 100 days, 20 have been rainy
and that on those days the number of gallons needed was as follows: there was one day
when 6 gallons were needed, one day when 9 gallons were needed and one day when 10
gallons were needed. There were two days when 7 gallons were needed and 15 where
8 gallons were needed. (Use this data as if it provides exact probabilities, rather than
estimates.)
3. Answer the following statements TRUE or FALSE, providing a succinct explanation
of your reasoning.
(a) You roll two fair three-sided dice. The probability the two dice show the same
number is 1/3.
(b) If events A and B are independent and P(A) > 0 and P(B) > 0, then P(A and B) >
0.
(c) If two events A and B are independent and P(A) > 0 and P(B) > 0, then A and
B cannot be mutually exclusive.
(d) If P(A and B) ≥ 0.4 then P (A) ≤ 0.40.
4. Prove the following properties of expectation directly from the definition. You may
assume a finite, discrete sample space for simplicity.
(a) E(aX) = aE(X)
(b) E(a + X) = a + E(X)
(c) E(X + Y) = E(X) + E(Y)
5. Come up with a counterexample to show that E(XY ) 6= E(X)E(Y ).
6. Show that, if X ⊥
⊥ Y , then E(XY ) = E(X)E(X).
7. Prove the following properties of variance in the finite, discrete sample space case.
1
, (a) V(aX) = a2 V(X)
(b) V(a + X) = V(X)
(c) V(aX + bY ) = a2 V(X) + b2 V(Y ) + 2abCov(X, Y )
8. Derive the identity: V (X) = E(X 2 ) − E(X)2 .
p
9. Consider the random variable Z = (X − E(X))/ (V(X)). Show that E(Z) = 0 and
V(Z) = 1.
10. Show that if two random variables are independent, then the correlation between them
is zero. Then, construct a counter-example to show that two dependent random vari-
ables can have zero correlation.
11. Show that E(X) = E(E(X | Y )) where the inner expectation is with respect to the
random variable X for a fixed value of Y and the outer expectation is over the random
variable Y . That is, show that
( )
X X
E(X) = p(y) x p(x | y) .
y x
This is called the iterated expectation formula.
12. An oil company wants to drill in a new location. A preliminary geological study
suggests that there is a 20% chance of finding a small amount of oil, a 50% chance
of a moderate amount and a 30% chance of a large amount of oil. The company has
a choice of either a standard drill that simply burrows deep into the earth or a more
sophisticated drill that is capable of horizontal drilling and can therefore extract more
but is far more expensive. The following table provides the payoff table in millions of
dollars under different states of the world and drilling conditions
small medium large
standard $20M $30M $50M
horizontal -$20M $40M $90M
(a) Find the mean payoffs of the two different drilling strategies.
(b) Find the variance in payoffs of each strategy.
(c) Which strategy would you advocate for and why?
(d) How much are you willing to pay for a geological evaluation that would determine
with certainty the quantity of oil at the site prior to drilling?
13. In a sample of 100,000 emails you find that 550 are spam. Your next email contains the
word “bigger”. From historical experience, you know half of all spam email contains
the word “bigger” and only 2% of non-spam emails contain it. Find the probability
that this new email is spam.
2
