Bus152 problem set 3 solutions

BUS 152     PROBLEM SET 3 – SOLUTIONS

SOME BASIC DISCRETE PROBABILITY DISTRIBUTIONS

Problem 1 (5.1)

Given the following probability distribution:

 Distribution A Distribution B X P(X) X P(X) 0 0.50 0 0.05 1 0.20 1 0.10 2 0.15 2 0.15 3 0.10 3 0.20 4 0.05 4 0.50

a. Compute the expected value for each distribution.

b. Compute the standard deviation for each distribution.

c. Compare and contrast the results of distributions A and B.

Problem 2 (5.3)

Using the company records for the past 500 working days, the manager of Konig Motors, a suburban automobile dealership, has summarized the number of cars sold per day into the following table:

 Number of Cars Sold per Day Frequency of Occurrence 0 40 1 100 2 142 3 66 4 36 5 30 6 26 7 20 8 16 9 14 10 8 11 2 Total 500

a. Form the probability distribution for the number of cars sold per day?

b. Compute the mean or expected number of cars sold per day?

c. Compute the standard deviation?

Problem 3

Using the local branch office records of a bank for the past 2 years (520 weeks), the manager of the bank has summarized the number of mortgages approved per week into the following table.

 Approved Home Mortgages per Week Frequency of Occurrence 0 52 1 104 2 208 3 78 4 52 5 26 Total 520

a. Form the probability distribution for the number of approved mortgages per week?

b. Compute the mean or expected number of approved mortgages per week?

c. Compute the standard deviation?

Problem 4 (5.7)

Given the probability distributions for variables X and Y:

 P(XiYi) X Y 0.4 100 200 0.6 200 100

Compute

a. E(X) and E(Y).

b. sX and sY

c. sXY

d. E(X+Y)

Problem 5 (5.9)

Two investments X and Y have the following characteristics:

 E(X) = \$50 E(Y) = \$100 s2X = 9,000 s2Y = 15,000 sXY = 7,500

If the weight assign to investment X of portfolio assets is 0.4, compute the

a. portfolio expected return,

b. portfolio risk.

Problem 6 (5.12)

You are trying to develop a strategy for investing in two different stocks. The anticipated annual return for a \$1,000 investment in each stock has the following probability distribution:

 Return (\$) Probability Stock X Stock Y 0.1 -100 50 0.3 0 150 0.3 80 -20 0.3 150 -100

Compute the

a. expected return for stock X and for stock Y.

b. standard deviation for stock X and for stock Y.

c. covariance of stock X and stock Y.

d. Would you invest in stock X or stock Y? Explain

Problem 7 (5.13)

Suppose that in the above problem (Problem 6) you want to create a portfolio that consists of stock X and stock Y. Compute the portfolio expected return and portfolio risk for each of the following percentages invested in stock X:

a. 30%,

b. 50%,

c. 70%.

d. On the basis of the results of (a) through (c), which portfolio would you recommend? Explain

Problem 8 (5.19)

If n = 5 and p = 0.40, what is the probability that

a. X = 4,

b. X £ 3,

c. X < 2,

d. X > 1.

Problem 9 (5.21)

The increase or decrease in the price of a stock between the beginning and the end of a trading day is assumed to be an equally likely random event. What is the probability that a stock will show an increase in its closing price in five consecutive days?

Problem 10 (5.22)

Sixty percent of Americans read their employment contracts, including the fine print. Assume that the number of employees who read every word of their contract can be modeled using the binomial distribution. For a group of five employees, what is the probability that

a. all five will have read every word of their contract?

b. at least three will have read every word of their contract?

c. less than two will have read every word of their contract?

Problem 11 (5.23)

A student is taking a multiple-choice exam in which each question has four choices. Assuming that she has no knowledge of the correct answer to any of the questions, she has decided on a strategy in which she will place four balls marked A, B, C, and D into a box. She randomly selects one ball for each question and replaces the ball in the box. The marking on the ball will determine her answer to the question. There are five multiple-choice questions on the exam. What is the probability that she will get

a. five questions correct?

b. at least four questions correct?

c. no questions correct?

d. no more than two questions correct?

Problem 12 (5.27)

When a customer places an order with Rudy’s On-Line Office Supplies, a computerized accounting information system (AIS) automatically checks to see if the customer has exceeded his or her credit limit. Past records indicate that the probability of customers exceeding their credit limit is 0.05. Suppose that, on a given day, 20 customers place orders. Assume that the number of customers that the AIS detects as having exceeded their credit limit is distributed as a binomial random variable.

a. What are the mean and standard deviation of the number of customers exceeding credit limit?

b. What is the probability that zero customers will exceed their limit?

c. What is the probability that one customer will exceed his or her limit?

d. What is the probability that two or more customers will exceed their limits?

Problem 13 (5.31)

Assume a Poisson distribution.

a. If l = 2.0, find P(X ³ 2).

b. If l = 8.0, find P(X ³ 3).

c. If l = 0.5, find P(X £ 1).

d. If l = 4.0, find P(X ³ 1).

e. If l = 5.0, find P(X £ 3).

Problem 14 (5.33)

Assume that the number of network errors experimented in a day on a local area network (LAN) is distributed as a Poisson random variable. The mean number of network errors experienced in a day is 2.4. What is the probability that in a given day

a. zero network errors will occur?

b. exactly one network error will occur?

c. two or more network errors will occur?

d. fewer than three network errors will occur?

Problem 15 (5.37)

The U.S. Department of Transportation maintains statistics for mishandled bags per 1,000 passengers. In 2003 Delta had 3.84 mishandled bags per 1,000 passengers. What is the probability that in the next 1,000 passengers Delta will have

a. no mishandled bags?

b. at least one mishandled bag?

c. at least two mishandled bags?

Problem 16 (5.41)

In 2004, both Lexus and Kia improved their performance. American’s Lexus had 0.87 problems per car and Korean’s Kia had 1.53 problems per car. If you purchased a 2004 Lexus, what is the probability that the car will have:

a. zero problems?

b. two or fewer problems?

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