251y9872 11/27/98 ECO251 QBA1 Name ______

THIRD EXAM Section Enrolled: MWF TR 10 11 12:30

NOVEMBER 24, 1998

Part I. Do all the Following (10+ Points) Penalty for not doing question 1. Show your work!

You are following a stock and wish to compare its return to the Dow-Jones index. The return for 9 periods on the stock () is compared below to the return on “buying the Dow”().

Period / (Dow) / (Stock) / /
1 / 12 / 12 / 144 / 144
2 / 6 / 15 / 225 / 90
3 / 2 / -4 / 16 / -8
4 / 4 / 1 / 1 / 4
5 / 4 / 2 / 4 / 8
6 / 5 / -1 / 1 / -5
7 / 6 / -8 / 64 / -48
8 / -6 / -2 / 4 / 12
9 / 3 / 3 / 9 / 9
Total / 36 / 18 / 468 / 206

Note that

Compute the following:

  1. The sample variance for the stock (4).
  2. The sample covariance between the stock and the Dow (2)
  3. The sample correlation between the stock and the Dow (2)
  4. Interpret the correlation (1)

Answers to questions 5) and 6) must be based on the results in questions 1-4. Do not recompute the answers after changing !

  1. If all the numbers in the column were higher by 3 (i.e. 15, 18, -1, 4 etc.), what would the variance, covariance and correlation computed above be? (1.5)
  2. If all the numbers in the column were twice as high (i.e. 24, 30, -8, 2 etc.), what would the variance, covariance and correlation computed above be? (1.5)

Solution:

1) , ()

2), .

3)

4)The positive sign of , the sample correlation, indicates that x and y tend to move together. If we square , we get approximately .23, which on a zero to one scale indicates a relatively weak relationship.

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5)We are leaving alone, but replacing by . From the syllabus supplement and the outline if so that and

a)

b) If , and and .

c)

6)We are leaving alone, but replacing by . From the syllabus supplement and the outline if so that and

a)

b) If , and and .

c)

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Part II. Do the following problems ( do at least 40 points ). Show your work! Note: You only need 40 points out of the 72 below to get an A+ -. Do the parts that look easiest!

  1. The following table represents the joint probability of and .

Event 0 1 2

2 .25 .10 .05 .40 .80 1.60

3 .05 .25 .05 .35 1.05 3.15

4 .15 .10 ? .25 1.00 4.00

.45 .45 .10 1.00 2.85 8.75 So and

0+ .45 + .20 = 0.65

0+ .45 + .40 = 0.85

  1. Fill in the missing number. (1)
  2. Are and independent? Why?(2)
  3. Compute , the covariance of and , and interpret it. (3)
  4. Compute , the correlation of and , and interpret it. (3)
  5. Find the probability that is less than 6. (1)
  6. (i) Find the distribution of . (2)

(ii) Using only the results of a)-d), find the mean and variance of .(3)

Solution: a) Since the table must total 1.00, the missing number is 0.

b) and are independent if . In the lower right corner ,

so and are not independent .

c)

= 0(2)(.25) + 1(2)(.10) + 2(2)(.05)

+ 0(3)(.05) + 1(3)(.25) + 2(3)(.05)Negative, so x and y move in opposite

+ 0(4)(.15) + 1(4)(.10) + 2(4)(0) = 1.85directions.

d)

We measure the strength of a correlation by squaring it. If we square -.0048, we get .000023. On a zero to one scale, this is tiny, so correlation is very weak.

e)See below. Since the sums of x and y are all below 6,

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f-i)

The tables above show the sums of and and their probabilities . If we add the probabilities we find the distribution below.

2 / / .25 = / .25 / .50 / 1.00
3 / / .05 +.10 = / .15 / .45 / 1.35
4 / / .15 +.25+.05 = / .45 / 1.80 / 7.20
5 / / .10 +.05 = / .15 / .75 / 3.75
6 / / 0 = / .00 / .00 / 0.00
1.00 / 3.50 / 13.30

Only the and columns are needed. From this, and . But this is not how I asked you to compute and .

f-ii)


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  1. I sell both coal and oil. Although the prices of oil and coal fluctuate, I maintain a constant markup of $16.00 per ton for coal and $0.06 per gallon on oil. My fixed costs for coal storage are $600 a month. My fixed costs for oil storage are $300/month.

My mean sales of coal are 3000 tons, with a standard deviation of 50 tons.

My mean sales of oil are 4500 gallons with a standard deviation of 300 gallons.

The correlation between sales of coal and oil is –0.3.

  1. What are the mean and standard deviation of profits on coal? (3)
  2. What are the mean and standard deviation of profits on oil? (1)
  1. What is the covariance between sales of coal and oil? (1.5)
  2. What is the covariance between profits on coal and oil? (1.5)
  3. What is the mean and standard deviation of total profits (oil profits plus coal profits)? (4)

Solution: If coal is and oil is , and .

a)Let profits on coal be. From the syllabus supplement and the outline if so that and , then:

or

b)Let profits on oil be . From the syllabus supplement and the outline if so that and , then:

or .

c)

d)

e)From the previous page substituting for and for ,

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  1. Assume that my father gives me a present of $1000 every time my wife has a boy baby and $2000 every time she has a girl baby. (Assume that the chance of a girl baby is 50%)
  1. What is the chance that out of 10 births exactly 3 will be girls? (2)
  2. What is the chance that out of 10 births more than half will be girls? (2)
  3. What is the mean and variance for the amount of money that I receive on any one birth? (2)
  4. What is the mean and variance for the number of girls born in 10 births? (1.5)
  5. What is the chance that the first girl will be born on the 10th birth ? (1.5)
  1. If a variable has the binomial distribution with and n=10

(i)Find the probability that you have at least one success (2)

(ii)Is it appropriate to use the Poisson distribution in this problem? Why? If it is appropriate, use the Poisson distribution to find the answer to (i). (2)

Solution:

Binomial . (Because we are asking for the probability of successes in tries, and the probability of success on any one try is given and constant.)

a), also .

b)

c) So

1000 .5 500 500000

2000 .510002000000

1.015002500000

d)

e) Geometric (Because we are asking for the probability of the first success on try.)

f) (i) Binomial .

(ii) Since is less than 500, the Poisson distribution is not appropriate.
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4 . . Make sure that you state clearly what distribution you are using in each section of this problem.

If I find that my paint jobs have an average of 0.10 blisters per square foot and a refrigerator has 14 square feet of painted surface,

  1. What is the chance that there will be exactly one blister on a refrigerator? (1)
  2. What is the chance that there will be at least one blister on a refrigerator? (2)
  3. What is the standard deviation of the number of blisters on a refrigerator? (1)
  4. Using your results from a), what is the chance that, in a delivery of 10 refrigerators, all will have at least one blister? (2)
  5. In d) what is the chance that more than 8 will have at least on blister? (2)
  6. If a store sells an average of 72 refrigerators in a 16 hour day and it takes 2 hours to get a delivery, what is the fewest number of refrigerators it should have in stock before it reorders if it wishes to limit the chance of running out to not more than 1% ? Explain. (3)

Solution:

Poisson with parameter (mean) of 0.1(14) = 1.4. (Because we are asking for the probability of successes when the average number successes in a unit of space or time is given and constant.)

a)

b)Poisson

c)Since, for the Poisson distribution, the mean and variance are identical,

d)Binomial . (Because we are asking for the probability of successes in tries, and the probability of success on any one try is given and constant.) so

e)Binomial Since,

f)Since there are 8 two-hour periods in a 16 hour day, this is a Poisson problem with . From the Poisson (9) table, the first value of with is 17 with . So the reorder point is 15.

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5.Assume that in any batch of soft drink cans 5% are defective. Make sure that you state clearly what distribution you are using in each section of this problem.

  1. If I take 4 cans from a package of 20 cans what is the chance that exactly 2 are defective? (2)
  2. If I take 4 cans from a package of 20 cans, what is the chance that at least one is defective? (2)
  3. What are the mean and variance of the number of defective cans in a sample of 4 taken from a package of 20? (2)
  4. If I take 4 cans from a package of 2000 cans, what is the chance that exactly 2 are defective? (2)
  5. If I take 4 cans from a package of 2000 cans, what is the chance that at least one is defective? (2)
  6. What are the mean and variance of the number of defective cans in a sample of 4 taken from a package of 2000? (2)
  7. If I am taking cans from a package of 2000, what is the chance that I will find my first defective can in the first 100 cans (2) ?

Solution: Hypergeometric with (Because we are asking for the probability of successes when the population is of limited size and the number of successes in the population is also limited.)

a)Since

b)Since ,

c)

d)Though hypergeometric is not wrong here, because of the large population we are better off using the binomial. So the distribution is either Binomial with or Hypergeometric with . or

e)or

f)

g)Geometric (Because we are asking for the probability of the first success on try.) Remember where .

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6.A jorcillator has two components, a phillinx and a flubberall. As long as both are working the jorcillator will not be junked. If either fails the jorcillator must be junked.

  1. If a phillinx has a lifespan approximated by a uniform distribution between 4 and 14 years, find:

(i)The mean and standard deviation of its life. (2)

(ii)The probability that it will last between 0 and 5 years (2)

(iii)The probability that it will last between 5 and 10 years (1)

(iv)The probability that it will last more than 10 years (1)

  1. If a flubberall has a lifespan approximated by a uniform distribution between 5 and 20 years, do (i) a(ii)-a(iv) above for the flubberall (2) and then (assuming that the lifespan of the two components is independent) find:

(ii)The probability that the jorcillator will last between 0 and 5 years. (2)

(iii)The probability that the jorcillator will last between 5 and 10 years. (3)

(iv)The probability that the jorcillator will last more than 10 years. (2)

Solution:

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a)For the Phillinx:

(i)

For the logic of the rest of this problem, see problem H4.

(ii) =.100

(iii) =.500

(iv) =.400

Phillinx (Draw a diagram for the uniform distribution)

Flubberall (Draw a diagram for the uniform distribution)

b)For the Flubberall:

(i)

=.3333 =.6667

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b-ii) A joint probability table follows on the left with the unions of joint events that make up the events that were specified summed to the right. Note that the probabilities sum to one.

/ Event / Component Joint Events / Probability
Fails in
0-5 / / /
Fails in
5-10 / / /
Fails in
10+ / / /
Sum / 1.0000

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