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ECO251 QBA1
THIRD EXAM
Version 1, Apr 20, 2006
Name: _____________________
Student Number: _____________________
Class Time (Circle) MWF 1, MWF 2, TR 12:30, TR 2.
Part I: 16 points.
z follows the standardized Normal distribution
Find the following. Make diagrams!
1. P3.11  z  2.00 
2. P3.02  z  3.02 
3. F 1.25   Pz  1.25 
4. z .175
z ~ N 0,1 .
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x follows the Normal distribution x ~ N 2, 7 .
Find the following. Make diagrams!
5. P3.11  x  2.00 
6. P3.02  x  3.02 
7. F 1.25   Px  1.25 
8. x.175
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Part II: (15+ points) Do all the following: All questions are 2 points each except as marked. Exam is
normed on 50 points including take-home. (Showing your work can give partial credit on some
problems! In open-ended questions it is expected. Please indicate clearly what sections of the
problem you are answering and what formulas you are using. Neatness counts!) Remember that you
may not be able to finish this section, so ration your time on each problem. [Numbers in brackets are a
cumulative total]
1.
2.
3.
If x is binomial, and n  20 , find the following (If you substitute another distribution for the
binomial, justify it!):
a) P3  x  9 when p  .40 . (1.5)
b)
P10  x  16  when p  .75 . (1.5)
c)
P1  x  5 when p  .03 . (2)
d)
Px  1 when p  .23 . (2)
A student calculates that P10  x  35   F 35   F 9 . The student could be working with the
a) Binomial Distribution
b) Normal distribution.
c) Continuous Uniform Distribution
d) Exponential distribution
e) All of the above
f) None of the above.
[9]
If  xy  .4 ,  x  3.2 ,  y  7.5 , w  7 x  5 and v  3 y  4 , find the following:
a)  xy (1)
b)  x y (1.5)
c)  wv (1.5)
d)  w (1.5)
[14.5]
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4.
What type of probability distribution will most likely be used to analyze the number of cars with
old model radios in the following problem?
From an inventory of 48 new cars being shipped to local dealerships, corporate reports
indicate that 12 have old model radios installed. The sales manager of one dealership wants to
predict the probability out of the 8 new cars it just received that, when each is checked, no
more than 2 of the cars have old model radios.
a) binomial distribution.
b) Poisson distribution.
c) hypergeometric distribution.
d) none of the above.
[16.5]
5.
The West Nantmeal police department must write, on average, 5 tickets a day to keep department
revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson
distribution with a parameter of 6.5 tickets per day. Interpret the value of the parameter. (1.5)
a) The number of tickets that is written most often is 6.5 tickets per day.
b) Half of the days have less than 6.5 tickets written and half of the days have more than 6.5
tickets written.
c) If we sampled all days, the arithmetic average or expected number of tickets written
would be 6.5 tickets per day.
d) The mean has no interpretation since 6.5 tickets can never be written. [16]
6.
(Extra credit ) Ben Horim and Levy tell us about a presidential debate in which the sound system
failed for a half hour. Assume that the sound system has a failure rate of 1 in 10 hours, which
means that the average wait for a failure is 10 hours. Use the exponential distribution.
a) What is the probability that the system will fail during a 3-hour event? (1.5)
b) What is the probability, that if a back-up system with the same characteristics as the
original system is used during the half hour that the original system is down for repairs, it
will fail before the half hour elapses?. (1.5)
c)
What is the variance of the elapsed time before a failure? (1)
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7.
Using the joint probability table below, (a) Fill in the missing number (1) (b) check for
independence (1), (c) Compute Covx, y  , Corr x, y  (4), (d) Compute Ex  y  and Var x  y  .


(2) (e) Find Px  y  4 and P x  y  4 x  0 (2).
y
0
 2 .1

x
4 0
5 .1

2 4
0 .1

.1

0 .1

[26]
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(Blank page for calculations)
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ECO251 QBA1
THIRD EXAM
Apr 25, 2005
TAKE HOME SECTION
Name: _________________________
Student Number: _________________________
Throughout this exam show your work! Please indicate clearly what sections of the problem you are
answering and what formulas you are using.
Part III. Do all the Following (19+ Points) Show your work! Neatness counts!
1. Ben Horim and Levy present the following actual returns for stock in Republic Steel and General Foods
over a ten year period.
Year
R1
1
22.5
2
9.3
3 -18.6
4
48.1
5
44.4
6
56.0
7 -20.1
8 -11.0
9
17.1
10
22.3
R2
-23.7
8.6
4.3
-52.2
62.4
61.4
9.8
52.6
17.1
11.7
Before you start, personalize the data below as follows. Take the third to last digit of your student number
and subtract it from -52.7 in the R2 column. This will make the results for year 4 even more disappointing
than they actually were. (Example: Seymour Butz’s student number is 123450. He subtracts the 4 from 52.7 getting -56.7).
a) Find the sample mean, variance and coefficient of variation for R2. Use computational formulas.
The results must look as if you calculated them by hand, although it is perfectly reasonable to
check the results using a computer or a calculator. (2.5)
b) Find the sample covariance and correlation between R1 and R2. (3)
c) Using a conventional measure of risk, explain which stock is riskier. (1).
[6.5]
d) Make a table showing 11 portfolios. Start with a portfolio with all your money invested in stock 1
(P1 = 1, P2 = 0), then move to a portfolio with 90% in stock 1 and 10% in stock 2 (P1 = .90, P2 =
.10), and move by tenths until you get a portfolio that is all stock 2. (5). The table should look as
P1
1
.9
.8
.7
.6
follows with much wider columns.
.5
.4
.3
.2
.1
0
R
s
C
Here R is the weighted average return,
s is the standard deviation of the portfolio return, and C is the coefficient of variation. [11.5]
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e)
f)
Make a graph . Put the mean return on the y axis and the standard deviation on the x axis. (3)
Connect your points in a curve. The C-like curve you will probably get illustrates the tradeoffs
between return and risk and is called an investment opportunities frontier.
Looking at the graph and the coefficients, are there stock portfolios that you would never
recommend? Why? What portfolio would you recommend to an 80-year old widow? Why? What
portfolio would you pick for yourself? Why? (1.5)
[16]
2. It’s time for a jorcillator problem! As everyone knows a jorcillator has two components, a Phillinx and a
Flubberall.
The life of a Phillinx, x , is described by a continuous uniform distribution between c  0  h5 and
d  30  h6 , where h5 and h6 are the last two digits of your student number.
The life of a Flubberall, y , is described by a Normal distribution with a mean of   20  h5 and a
standard deviation of   5.10 .
(Example: Seymour Butz’s student number is 123450. c  0  5  5 , d  30  0  30 and   20  5  25 ).
a)
Find the following probabilities for the Phillinx: (i) PPH 1   P0  x  10  , the probability that it
dies before the end of the tenth year, (ii) PPH 2   P10  x  20  , the probability that it dies
between the end of the tenth and the end of the twentieth year and (iii) PPH 3   P( x  20) , the
probability that it lasts beyond the twentieth year. (1.5)
b) Find the following probabilities for the Flubberall: (i) PF1   P0  y  10  , the probability that it
dies before the end of the tenth year, (ii) PF2   P10  y  20  , the probability that it dies
between the end of the tenth and the end of the twentieth year and (iii) PF3   P( y  20 ) , the
probability that it lasts beyond the twentieth year. (3)
c) Make a joint probability table using the 6 events described above. Assume independence. (1)
d) Assume that the Jorcillator will fail only if both components fail. Find the following probabilities
for the Jorcillator: (i) PJ 1  , the probability that it dies before the end of the tenth year, (ii)
PJ 2  , the probability that it dies between the end of the tenth and the end of the twentieth year
and (iii) PJ 3  , the probability that it lasts beyond the twentieth year. (3)
[23.5]
e)




(Extra credit) Find the following conditional probabilities: : (i) P J 3 F1 , (ii) P J 3 F2 (iii)
PJ 1 F3  , PF1 J 2  (2).
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