CLO 1: Applv econometric and statistical theories and tools to analvze business problems on

UAE/GCC data

Problem 1:

You are organizing a cultural event at Dubai Media Center next Saturday (20th September 2014). You

believe that attendance will depend on the weather on that day, You consider the following possibilities

are appropriate: (1+1+1+1=4 Marks)

Weather Probability = f(x) Attendance (X)

Terrible weather 0.2 500

Mediocre weather 0.6 1000

Great weather 0.2 2000

a. If X denotes the attendance, why is X a random variable?

b. What is the expected number of attendance?

c. Suppose that you will charge AED5 per attendee and the total cost of setting the event is

AED 2,000. What is your expected profit?

d. If the variance of attendance is cx2= 240,000, find the variance of your profit.

Problem 2:

As you walk into your Quantitative analysis exam, a friend bets you AEDIO that he will outscore you on

the exam. Let X be a random variable denoting your winnings. X can take the value 10 if you win, 0 if

there is a tie, or -10 if you loss. You know that the probability distribution for X, f(x), depends on whether

he studied for the exam or not. Let Y=O if your friend studied and Y=l if he did not study. Cor~siderth e

following joint probability distribution table: (1.5+1+1+1 = 4.5 Marks)

a. Fill in the missing elements in the table.

b. Compute E(X). Should you take the bet?

c. What is the probability distribution of your winning if you know that your friend did not study?

d. Find your expected winnings given that your friend did not study.

Problem 3:

Your firm’s marketing manager believes that total sales X can be modeled using a normal distribution with

mean p = AED 2.5 million and standard deviation o = AED 300,000. (1+0.5=1.5 Marks)

a. What is the probability that your firm’s sales will exceed AED 3 million?

b. Draw a sketch to illustrate your calculation.

Problem 4:

[Suppose] At Carrefow Dubai City Centre the sales of canned tuna varies from week to week. Marketing

researchers have determined that there is a relationship between sales of canned tuna and the price of

tuna. Specially,

SALES ~40710- 430PRICE

where SALES are nurr~ber of cans sold per week and PRICE is measured ill cents per can. Suppose

PRICE over the year can be considered (approximately) a normal random variable with mean p 75

cents and standard deviation o = 5 cents. That is PRICE – N(75,25). (1 +1+1+1 = 4 Marks)

a. What is the numerical expected value of SALES?

b. What is the numerical value of the variance of SALES?

c. Find the probability that more than 6,300 cans are sold in a week.

d. Draw a sketch illustrating your calculation in part c.

Problem 5:

Let X be a discrete random variable that is the value shown on a single roll of a fair die.

(1+1+1+1++1+1.5 = 5.5 Marks)

a. Represent the probability density function f(x) in tabular form.

b. What is the probability that X=4? That X=4 or X=5?

c. Find the expected value of X2.

d. Find the variance of X.

e. Obtain a die. Roll it 20 times and record the values obtained. What is the average of the first 5

values?, What is the average of the first 10 values? What is the average of the first 20 values?

Problem 6:

Suppose that X and Y are random variables with expected values px = py = p and variances 02 = 02 =

02. Let Z= (X+Y)12. (1+1+1= 3 Marks)

a. Find the expected value of Z (i.e. E(Z)).

b. Find var(Z) assuming that X and Y are statistically independent.

c. Find var(Z) assuming that cov(X, Y) = 0.50~.

Problem 6:

Suppose Al Daman UAE Value Fund has an ar~nual rate of return that is approximately normally

distributed with mean (expected value) 5% and standard deviation 4%. Use Table 1 (the Cumulative

Probabilities for the Standard Normal distribution table at page 742 of the course textbook) to answer the

following questions: (1+1+1.5=3.5 Marks)

a. Find the probability that your I-year return will be negative.

b. Find the probability that your I-year return will exceed 15%.

c. If the mutual fund managers modify the composition of its portfolio, they can raise its mean

annual return to 7%, but will also raise the standard deviation of returns to 7%. Would you

advice the fund managers to make this portfolio change? Why?

Problem 7:

An investor holding a portfolio consisting of two stocks invests 25% of assets in Stock-A and 75% into

Stock-B. The return RA from Stock-A has a mean of 4% and a standard deviation of GA 8%. Stock-B

has an expected return E(RB) 8% with a standard deviation of GB = 12%. The portfolio return is

P=O.~~R+ 0A.7 5Re. (1+1+1+1=4 Marks)

a. Compare the expected return on portfolio.

b. Compare the standard deviation of the returns on the portfolio assuming that the two stocks’

returns are perfectly positively correlated.

c. Compute the standard deviation of the returns on the portfolio assumiog that the two stocks’

returns have a correlation of 05.

d. Compute the standard deviation of the returns on the portfolio assuming that the tow stocks’

returns are uncorrelated.

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