MATH 218 FINAL EXAM
MAY 4, 2006
Problem 1
(25 pts)
.
A portfolio manager offers two assets
to potential investors.
Half of the investors select Asset A;
the remaining investors select Asset B. Asset A has three
possible returns:
8%, with a probability of 1/4; 12%, with
a probability of 1/2; 16%, with a probability of 1/4.
Asset
B also has three possible returns: 8%, with a probability of
1/3; 14%, with a probability of 1/3; 20%, with a probability
of 1/3.
(a)
Draw the appropriate probability tree. Be sure to in
clude the labels of events, probabilities, and condi
tional probabilities.
(b)
Find the probability that a randomly chosen investor
earns a return of at least 10%.
(c)
Given that a randomly chosen investor earns a return
of at least 10%, find the probability that the investor
selected Asset A.
(d)
Given that a randomly chosen investor earns a return
of at least 10%, find the probability that the return is
greater than 15%.
Problem 2
(20 pts)
.
The annual rate of return (in percent),
X
, of stock
A
has the following probability distribution:
x

10
0
10
20
30
P
(
X
=
x
)
0.17
0.20
0.26
0.20
0.17
For example,

10
represents a loss of 10% of the initial in
vestment, while
10
represents a gain of
10
%
of the initial in
vestment. After one year, an investor will hold the amount
of the initial investment plus any amount gained and minus
any amount lost.
(a)
Find the expected annual rate of return on stock A
(in percent).
(b)
Find the standard deviation of the annual rate of re
turn on stock A (in percent).
(c)
Jane starts with
$
1,000, borrows
$
200 at a constant
annual riskfree rate of
6
%
, and then invests the sum
of
$
1, 200
in stock A. After one year, Jane’s invest
ment will be worth
$
1200
(
1
+
0.01X
) 
$
200
(
1.06
) =
$
988
+
$
12X
.
Find the expected value of Jane’s investment (in dol
lars and cents) after one year.
(d)
Find the standard deviation of Jane’s investment (in
dollars and cents) after one year.
Problem 3
(20 pts)
.
The two common types of errors made
by programmers are syntax errors and logic errors.
For a
simple language such as BASIC the number of such errors is
usually small. Let
X
denote the number of logic errors and
Y
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 Fall '06
 Haskell
 Sets, Normal Distribution, Probability, Probability theory, probability density function

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