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These problems will NOT be collected or graded, but they will be useful for studying for exams.
1)
The true annual return on a $200 investment in Zipco stock is as follows:
Return in percent (X)
Return in dollars (Y)
Probability
3
%
$
6
.
4
6%
$12
.6
a) Find the mean of the random variable X __________ variance __________
and standard deviation________
b) What are the units?
c) Find the mean of the random variable Y __________ variance __________
and standard deviation________
d) What are the units?
2)
The true annual return on Pizco stock is identical to Zipco, and the returns are independent so the true linear correlation
between them = 0.
a) What is the average return (in %) for a portfolio of 50% Pizco and 50% Zipco? __________
b) Suppose your portfolio is $100 in Pizco and $100 in Zipco. What is the average return (in $)? ________
c) Consider the return in percent of this portfolio.
Find the variance __________
and standard deviation________
d) Consider the return in dollars of this portfolio.
Find the variance __________
and standard deviation________
e) Would an investor be better off investing $100 in each of these two stocks rather than $200 in just one (ignore transactions
costs of buying and/or selling stocks)?
Briefly explain your answer.
3)
a) Suppose the probability of Bob getting an “A” in any class is .2 (20%).
Assuming his grades in class are independent of
one another, what is the probability that he takes three classes and receives ALL As?
b) What is the probability that he takes three classes and receives NO As?
4)
You flip a fair coin (probability of head on any one toss) three times, and record whether a head or tail faced up on each of the
three tosses.
Each group of three tosses is one trial or experiment.
a) Write the
sample space
, showing all eight of the
equally likely outcomes
:
b) Let X = number of heads in three tosses.
Write the possible values of X with their associated probabilities:
X
Probability
c) What is the true population mean of X?
μ
X
=
_____
d) What is the true population variance of
X?
σ
X
2
=
_____
e) What is the true population standard deviation of X?
σ
X
=
_____
5)
Suppose you perform the experiment above four times (each single experiment is 3 tosses) and get the following results:
i) HHH;
ii) HTT;
iii) THH;
and iv) THH
a) What is the sample mean of X?
=
X
_______
b) What is the sample standard deviation of X?
s
X
= _______
c) Draw a
histogram
showing your actual outcomes below to the left, with X values on the horizontal axis and relative
frequency of X on the vertical axis.
d) Draw the true
probability histogram
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 Spring '07
 Guggenberger

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