Risk and Uncertainty
[ET 2031]
Last name
1 February 2017
First name
Matriculation
Instructions. The time allotted for the exam is 1h55’. The exam is closed-book and closednotes. You are allowed to use standard tables for the main distributions, a pocket calculator
and two sides of A4-paper prepared by you. (Any electronic device, including mobile phones,
must be offline.) Exchanging information or communicating with other people, as well as any
form of cheating, implies the immediate disqualification of your exam. There is a total of 11
independent questions, that may have different degrees of difficulty. Each question is worth 3
points. You are asked to solve all questions. For numerical answers, use three significant digits
and reduce fractions to lowest terms. This exam has seven pages.
Please write first and last name as well as your matriculation number on top of
this page. Make sure to return all stapled pages, and nothing else.
Sign below to confirm that, to the best of your knowledge, you are clear of didactic debts or
any other circumstances that may invalidate your exam:
If you wish to withdraw from the exam, please sign here:
The section below is reserved for grading
Question
1
2
3
4
5
6
Total
Points
6
6
6
6
6
3
33
Scores
1. (a) An elevator in a building starts with five passengers and stops at seven floors. If
every passenger is equally likely to get off at each floor and all the passengers leave
independently of each other, what is the probability that no two passengers will get
off at the same floor?
(b) You have to play Alekhine, Botvinnik, and Capablanca once each. You win each game
with respective probabilities pa = 0.4, pb = 0.3, pc = 0.2. You win the tournament
if you win two consecutive games (otherwise you lose), and you can choose in which
order to play the three games.
Show that you maximize your chance of winning if you play Alekhine second.
(a) There are 75 possible outcomes. The probability is
7·6·5·4·3
360
=
≈ 0.1499
5
7
2401
(b) If you play in an arbitrary order 1-2-3, the probability of winning is
p1 p2 + (1 − p1 )p2 p3 = p2 [p1 + p3 − p1 p3 ]
The effect of the first and third opponent is symmetric so it suffices to consider three
cases depending on whom you play second. With Alekhine second, the probability of
winning is 0.176. With Botvinnik second, it is 0.156. With Capablanca, it is 0.116.
Intuitively, you want to have the player you are most likely to beat in the second
place because winning the second match is a necessary condition for winning the
tournament.
2. (a) Each time a shopper purchases a tube of toothpaste, he chooses either brand A
or brand B. Suppose that for each purchase after the first, the probability is 1/3
that he will choose the same brand that he chose on his preceding purchase and the
probability is 2/3 that he will switch brands. If he is equally likely to choose either
brand A or brand B on his first purchase, what is the probability that both his first
and second purchases will be brand A and both his third and fourth purchases will
be brand B?
(b) Suppose that 88 percent of all economics students are shy, whereas only 22 percent
of all business students are shy. Suppose also that 90 percent of the people at a large
gathering are economics students and the other 10 percent are business students. If
you meet a shy person at random at the gathering, what is the probability that the
person is a business student?
(a) The probability is 1/27.
(b) The probability is
P (B|S) =
.22 × .10
≈ 2.71%
.22 × .10 + .88 × .90
3. (a) Each printed character in a book is misprinted independently with probability p =
0.0002, or is correct with probability 1 − p. Let n = 88,000 be the number of
characters in the book, and let X be the number of misprinted characters. How is
X distributed? What is the expected number of misprints in the book?
(b) Suppose that a point in the xy-plane is chosen at random from the interior of a circle
for which the equation is x2 + y 2 = 1; and suppose that the probability that the
point will belong to each region inside the circle is proportional to the area of that
region. Let Z denote a random variable representing the distance from the center of
the circle to the point. Find the c.d.f. of Z.
(a) X has a binomial distribution. Hence, E(X) = np = 17.6.
(b) The distribution function for Z is

 0 if z < 0
z 2 if 0 ≤ z ≤ 1
FZ (z) =

1 if z > 1
4. (a) Suppose that the joint p.d.f. of two random variables X and Y is as follows:
c(x + y 2 ) for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1
f (x,y) =
0
otherwise
Determine the conditional p.d.f. of X for every given value of Y in [0,1], and find
Pr (X < 1/2|Y = 1/2).
(b) Suppose that a sequence of independent tosses are made with a coin for which the
probability of obtaining a head on each given toss is 0.25.
What are the expected value and the variance of the number of tails that will be
obtained before four heads have been obtained?
(a) The conditional p.d.f. is
f (x|y) =
x + y2
1/2 + y 2
and Pr (X < 1/2|Y = 1/2) = 1/3.
(b) This situation is modelled by a negative binomial with parameters r = 4 and p = 0.25.
Hence,
4 × 0.75
4 × 0.75
E(X) =
= 12 and V (X) =
= 48
0.25
0.252
5. (a) Suppose that X and Y are negatively correlated. Is V (X + Y ) larger or smaller than
V (X − Y )?
(b) Find the 0.50, 0.20, and 0.90 quantiles of the standard normal distribution.
(a) If X and Y are negatively correlated, then Cov(X,Y ) < 0. Hence, V (X + Y ) =
V (X) + V (Y ) + 2Cov(X,Y ) ≤ V (X) + V (Y ) − 2Cov(X,Y ) = V (X − Y ).
(b) They are 0.00, about −0.84, and about 1.28.
6. Suppose that a person has a given fortune M > 0 and can bet any amount b of this fortune
in a game (0 ≤ b ≤ M ). If he wins the bet, then his fortune becomes M + b; if he loses
the bet, then his fortune becomes M − b. In general, let X denote his fortune after he
has won or lost. Assume that the probability of his winning is p and the probability of
his losing is 1 − p, with 0 < p < 1.
Assume that his utility function, as a function of his final fortune x, is U (x) = log x for
x > 0. If the person wishes to bet an amount b for which the expected utility of his
fortune E[U (X)] will be a maximum, what amount b should he bet?
The optimal amount is b∗ = (2p − 1)M if p ≥ 1/2 and 0 if p < 1/2.