OPIM Assignment 4
1. Cu = 24-11 = $13
Co = 11-7 = $4
Critical ratio = 13/(13+4) = 0.7647
μ = 30,000
σ = 10,000
Using normal distribution function (=norminv(0.7647,30000,10000)), the optimum order quantity is 37,216 jerseys to maximize profit.

2. Quantity = 32,000
First, we normalize the order quantity to find the z-statistic z=Q-μσ=32,000-25,00010,000=0.7
We then look up the standard normal loss function. The expected lost sale is given by. Lz=0.1429
Therefore, the expected lost sales = 10,000 * 0.1429 ≈ 1,429 Expected sales = 30,000 – 1,429 = 30,571 jerseys

3. First, we normalize the order quantity to find its z-statistic z=Q-μσ=28,000-20,00010,000=0.8
In-stock probability = normdist(0.8, 0, 1, 1) = 0.788145
Therefore, the probability of filling all demand is 0.788145

4. Quantity = 8,000
First, we normalize the order quantity to find the z-statistic z=Q-μσ=8,000-15,00010,000=-0.7
We then look up the standard normal loss function. The expected lost sale is given by. Lz=0.8429
Therefore, the expected lost sales = 10,000 * 0.8429 = 8,429 Expected sales = 15,000 – 8,429 = 6,571 jerseys
Expected leftover = 8000 – 6,571 = 1429 jerseys
Therefore, Nike has to sell 1,429 on discount

5. Cu = 16-11 = $5
Co = 9-7 = $2
Critical ratio = 5/(5+2) = 0.7143
μ = 40,000
σ = 10,000
Using normal distribution (=norminv(0.7143,40000,10000)), the optimum order quantity is 45660 jerseys in the first order to maximize profit.

6. Cu = 0.02-0.01 = $0.01
Co = 0.01-0 = $0.01
Critical ratio = 0.01/(0.01+0.01) = 0.5
μ = 300
σ = 75
Using normal distribution (=norminv(0.5,300,75)), the optimum order quantity is 300 million minutes from Vmail to minimize the expected capacity expense.

7. Q = 375
First, we normalize the order quantity to find the z-statistic z=Q-μσ=375-30075=1
We then look up the standard normal loss function. The expected lost sale is given by. Lz=0.0833
Therefore, the expected lost sales = 75 * 0.0833 = 6.2498...

...Unit 4ProblemSet 1: Normal Probability Distributions
4/27/2014
MA3110
Statistics
Otis Jackson
Unit 4problemset 1: Normal Probability Distributions
Page.285 Ex 6,8,10,12
6. x = 80, z=80-10015 = -1.33 z= 0.0918 1-0.0918 = 0.9082
8. x = 110, z=110-10015 = 0.67 z= 0.7486
z= 75-10015 = -1.67 z= 0.0475 0.7486-0.0475= 0.7011 (shaded area)
10. z= 0.84 (shaded) z= -0.84 x= 100+(-0.84∙15) = 87 (rounded)
12. . z= 2.33 x= 100+(2.33∙15) = 135 (rounded)
Page 288 Ex 34
34.Appendix B Data Set: Duration of Shuttle Flights
a. Find the mean and standard deviation, and verify that the data have a distribution that is roughly normal. Mean= 25317115 = 220.15 Standard Deviation=115253172-(25317)2115(115-1) = 86 (rounded)
The normal distribution is 115
b. Treat the statistics from part (a) as if they are population parameters and assume a normal distribution to find the values of the quartiles 1,2 and 3. Mean= 220.15 Standard Deviation= 86
Q1 = 220.5 + (-0.67 ∙ 86)= 162.53 Q2= 220.5 + (0.00 ∙ 86) = 220.5 Q3=220.5 + (0.67 ∙ 86) = 277.77
Page.300 Ex 20
Quality Control: Sampling Distribution of Proportion after constructing a new manufacturing machine. 5 prototype integrated circuit chips are produced and it is found that 2 are defective (D) and 3 are acceptable (A). Assume that two if the chips are randomly selected with replacement from this population
a. After identifying the 25...

...Week 4 Homework
Solutions: ProblemSet4
1. Determining Profit or Loss from an Investment. Three years ago, you purchased 150 shares of IBM stock for $88 a share. Today, you sold your IBM stock for $103 a share. For this problem, ignore commissions that would be charged to buy and sell your IBM shares.
a. What is the amount of profit you earned on each share of IBM stock? The profit on each share of IBM stock was $15. $103 priced when each share was sold, $88 priced when each share was purchased = $15.
b. What is the total amount of profit for your IBM investment? The total profit for the IBM transaction was $2,250. $15 profit per share x 150 shares = $2,250.
2. Calculating Rate of Return. Assume that at the beginning of the year, you purchase an investment for
$8,000 that pays $100 annual income. Also assume the investment’s value has decreased to $7,400 by the end of the year.
a. What is the rate of return for this investment?
Step 1 subtract the investment’s initial value from the investment’s value at the end of the year.
$7,400 – $8,000 = $600 (negative)
Step 2 add the annual income and the amounts from Step 1.
$600 (negative) + $100 = $500 (negative)
Step 3: divide the total dollar amount of return (Step 2) by the original investment
$500 (negative) ÷ $8,000 = .0625 (negative) = 6.25% (negative)
b. Is the...

...FIN 350
Prof. Porter
ProblemSet4
1. Describe what happens to the total risk of a portfolio as the number of securities is increased. Differentiate between systematic risk and unsystematic risk and explain how total risk and systematic risk are measured.
As the number of securities increases, the total risk of the portfolio decreases. This decrease occurs due to the benefits of diversification which is the process of acquiring a portfolio of securities that have dissimilar risk-return characteristics in order to reduce overall portfolio risk. The total risk of a security or a portfolio is measured with the variance or standard deviations of returns (std dev. ^2 = variance). The larger the standard deviation, the greater the total risk and the more likely it is that you will have a large price move.
Unsystematic risk is the unique or security specific risks that tend to partially offset one another in a portfolio. /this could happen when the price of one stock in the portfolio goes down, the price of another tends to go up, which partially offsets the loss. As long as the returns of two securities are not perfectly, positively correlated, one can reduce total risk by combining securities in a portfolio. By adding securities to a portfolio, it is possible to eliminate unsystematic risk.
Systematic risk is also known as market risk or nondiversifiable risk. The risk tends to affect the entire market in a similar...

...PROBLEMSET4
1) Consider the following utility functions, where W is wealth:
(a) U (W ) = W 2
1
(b) U (W ) =
W
(c) U (W ) = −W
(d) U (W ) = W
(e) U (W ) = ln(W )
(f) U (W ) =
W 1−γ
, with γ = 2
1−γ
How likely are each of these functions to represent actual investor preferences? Why?
2) Suppose investors have preference described by the following utility function
with A > 0:
U = E(r) − 1 Aσ 2
2
Each investor has to choose between three portfolios with the following characteristics:
E(rA ) = 20%
σA = 20%
E(rB ) = 12%
σB = 22%
E(rC ) = 15%
σC = 28%
(a) Which portfolio would every investor pick and why?
(b) What utility would an investor with a risk aversion parameter, A, of 1
get from the three portfolios?
(c) What must be the risk aversion of an investor that is indiﬀerent between
picking portfolio B and portfolio C?
1
3) Consider an investment universe consisting of three assets with the following
characteristics:
E(r1 ) = 12%
E(r2 ) = 17%
E(r3 ) = 7%
σ1 = 25%
ρ1,2 = 0.5
σ2 = 30%
ρ1,3 = 0.25
ρ2,3 = 0.35
σ3 = 20%
(a) What is the expected return and standard deviation of an equally weighted
portfolio investing in all three assets?
(b) What would the diversiﬁcation beneﬁt be for an investor that shifted
her investment to the equally weighted portfolio from an investment
consisting only of asset 1?
(c) If choosing between investing all her capital...

...4. What combination of the two goods below allows you to maximize your utility with a budget constraint of $14? Show how you arrived at your conclusion in the space provided below. Place your final answers on the lines at the bottom of this page.
PRICE = $0.50 per pint
|Bottles of glue |Total Utility (Utils) |
|1 |15 |
|2 |23 |
|3 |30 |
|4 |35 |
|5 |38 |
|6 |40.5 |
PRICE = $2.00 per box
|Bales of hay |Total Utility (Utils) |
|1 |10 |
|2 |22 |
|3 |36 |
|4 |52 |
|5 |70 |
|6 |90 |
Bottles of glue: ___4______________________
Bales of hay: _____6_______________________
We need to determine the MU/dollar spent. First, you calculate marginal utilty as shown in the table. Then divide that by the price of each good. After that, look for the place where MU/$ for glue = MU/$ for hay while falling under the $14 budget...

...NAME : _________________________________ PROBLEMSET4
1. What combination of the two goods below allows you to maximize your utility with a budget constraint of $14? Show how you arrived at your conclusion in the space provided below. Place your final answers on the lines at the bottom of this page.
PRICE = $0.50 per pint
Pints of Butter Beer
Total Utility (Utils)
1
15
2
23
3
30
4
35
5
38
6
40.5
PRICE = $2.00 per box
Boxes of Bertie Botts Every-flavor Beans
Total Utility (Utils)
1
10
2
22
3
36
4
52
5
70
6
90
Pints of Butter Beer: _________________________
Boxes of Beans: ____________________________
2. Assume the following game is played one time only. Based on the information in the payoff matrix, PNC Bank and Citizens Bank are considering an implicit collusive agreement on interest rates. Payoffs to the two firms are represented in terms of profits in thousands of dollars:
Citizens Bank
Collude: Raise Rates
Defect: Keep Rates where they are
PNC
Collude: Raise Rates
(900, 600)
(700, 800)
Defect: Keep Rates where they are
(1100, 300)
(800,400)
a. Does PNC have a dominant strategy? What is it? Does Citizens have a dominant strategy? What is it?
b. Solve for the Nash equilibrium.
c. Does the result of your answer change if the game is played an infinite number of times? Why or why not. Properly use game theoretic terminology in your answer.
3....

...ECE 302
ProblemSet 9
Fall 2013
The following problems have been selected from the course text.
4.78 In a large collection of wires, the length of a wire is X, an exponential random variable with mean
5π cm. Each wire is cut to make rings of diameter 1 cm. Find the probability mass function for the
number of complete rings produced by each length of wire.
4.85 The exam grades in a certain class have a Gaussian pdf with mean m and standard deviation σ. Find
the constants a and b so that the random variable Y = aX + b has a Gaussian pdf with mean m and
standard deviation σ .
4.86, 4.87 Let X = U n where n is a positive integer and U is a uniform random variable in the unit interval.
Find the cdf and pdf of X. Repeat for the case where U is uniform in the interval [−1, 1].
4.94 modiﬁed Let Y = α tan(πX), where X is uniformly distributed in the interval (−1/2, 1/2).
a. Show that Y is a Cauchy random variable.
b. Find the pdf of Z = 1/Y .
4.96 Find the pdf of X = − ln(1 − U ), where U is a uniform random variable in (0, 1).
4.99 modiﬁed Let X be a random variable with mean m. Compare the Chebyshev inequality and the
exact probability for the event {|X − m| > c} as a function of c for the case where:
a. X is a uniform random variable in the interval [−b, b];
b. X has pdf fX (x) =
α
2
exp(−α|x|);
c. X is a zero mean Gaussian random variable with variance σ 2 .
4.100 Let X be the number of successes in n...

...Me1
ProblemSet #2
The US College Enrollment and the “Third Law of Demand”
A theorem proposed by Professors Alchian and Allen in their text, University Economics (1964) has had several rebirths of interest in the literature. The so-called “third law of demand,” or “relative price theorem,” holds that a fixed cost added to a good of varying quality causes the consumer to prefer the category considered of higher quality to the lower.
Recently a number of studies, keeping this theorem in mind have looked into a relationship between the ratio of public to private enrollment and unemployment in cross-sectional as well as in time series data. Part of the full cost of participating in higher education is foregone employment income. In their regression model, these studies have regressed the public/private ENROLLMENT RATIO (as an indicator of relevant demand) against UNEMPLOYMENT RATES (as an indicator of cost) as well as a number of variables designed to account for “other things” which tend to vary at the same time, such as income, financial aid and tuition ratios. Tuition ratio is typically specified as the ratio of the full cost (including forgone employment income) of public higher education (Pa) to private higher education (Pb), where Pa is less than Pb.
In Table 1, below, a cross-sectional model reveals the relationships between relative education demands by public and private university students (as measured by state level ENROLLMENT...

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