Discrete Probability Distributions

1. For each of the following processes, identify the random variable and the outcome space.

a. An observer posted at a specified location on a lake counts the number of

Waterfowl within his field of view during a one-hour period.

Random variable- counts the number of waterfowl within his field of view during a one-hour period

Outcome space- waterfowl

b. A doctor reviews the outcome of a strep throat test.

Random variable- the outcome of a strep throat test

Outcome space- a strep throat test

c. An operator counts the number of vehicles that pass through a toll booth in an

eight-hour period.

Random variable- counts the number of vehicles that pass through a toll booth in an Eight-hour period

Outcome space- vehicles that pass through a toll booth

d. A candy company produces individually wrapped mints. The quality control

inspector removes and counts the pieces where the wrapping is not completely

sealed.

Random variable- counts the pieces where the wrapping is not completely

sealed.

Outcome space- the pieces where the wrapping is not completely

sealed.

2. The number of small cosmetic defects found in a manufactured product is given by the following probability distribution. Enter this probability distribution into a JMP datasheet. Create the necessary columns and formulas in JMP to calculate the

Average, variance, and standard deviation of the number of defects.

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Y P(Y=y) Variance

(P(Y=y)-average)2/5 Standard deviation

?variance

0 0.30 0.30-0.2 =(0.10)2=0.01

1 0.35 0.35-0.2 =(0.15)2=0.0225

2 0.25 0.25-0.2 =(0.05)2=0.0025

3 0.05 0.05-0.2 =(-0.15)2=0.0225

4 0.05 0.05-0.2 =(-0.15)2=0.0225

Average

=0.30+0.35+0.25+0.05+0.05

=1/5

=0.2 Variance = =0.01+0.0225+0.0025+0.0225+0.0225

5

=0.016 SD =?0.016

= 0.126

3. Calculate the mean, variance, and standard deviation of the following distribution of financial returns.

Economy Probability Function p(y) Return Y in 1000 $ y*p(y) (yMean[Y])2*p(y)

Recession 0.30 15 -4.5

Slow growth 0.40 15 6

Boom 0.30 30 9

SUM 1.0 n.a. Mean[Y]=10.5/3

= 3.5 V[Y]= SD[Y]=

Variance y*p(y)-average)2/3 = 33.5

SD= ?33.5= 5.79

4. An office supply retailer is planning the grand opening of a new store. Flyers containing information about the grand opening event and coupons for three complementary items are mailed to households within a 25-mile radius. The regional manager believes there is an equally likely chance that a customer will redeem none, one, two, or three coupons. Consider that the random variable Y = the number of coupons redeemed by a customer.

a. Sketch the probability function.

b. Sketch the cumulative distribution function.

c. What is the probability that a customer will redeem at least one coupon?

7. The joint probability distribution shown below quantifies a financial analysts belief

in the possible outcomes for two stocksone from the health care sector and the

other from the banking industry.

Returns from health care stock

Returns from bank stock 5 0 5

5 .15 .15 .05

0 .10 .20 0

5 .05 .05 .25

a. Set up a table to compute the means, variances, and covariance, and calculate the

quantities needed to determine the portfolio variance.

b. What are the expected returns and risk for each of the individual stocks?

c. Are the returns of these two stocks independent? Explain how you arrived at your

answer.

d. Compute the standard deviation for a portfolio with a 50% allocation of each

stock. How does it compare to the standard deviations of the two individual

stocks?

5.7 Case Study: Assessing Financial Investments

Background

In the context of investing, probability models can assist with the evaluation of risk and expected return. Investors often consider both the expected return and risk when making investment choices. The mean of the distribution is the expected return, which represents the return of the stock over a long time horizon. The standard deviation (square root of the variance) measures the risk associated with the stock. Lower variability in the returns indicates less risk as compared to a return with higher variability.

Business Problem

Two stocks are available in which to invest. The returns and associated probabilities for possible economic conditions are given in the following tables.

Returns from Woodside Corporation

Economic Condition Probability Function f(y) Return Y in %

Recession 0.1 10

No growth 0.2 1

Slow growth 0.5 5

Boom 0.2 12

Expected return = (0.1)(-10)+(0.2)(-1)+(0.5)(5)+(0.2)(12)= 3.7

risk Variance= 13.05

Standard deviation=?13.05=3.61

Returns from Brookside Corporation

Economic Condition Probability Function f(y) Return Y in %

Recession 0.1 2

No growth 0.2 0

Slow growth 0.5 3

Boom 0.2 6

Expected return =(0.1)(2)+(0.2)(0)+(0.5)(3)+(0.2)(6)=2.5

risk Variance= 3.73

Standard deviation=?3.73=1.93

Based on these probability distributions, find the expected return and risk for each of the two stocks. Compare their expected returns and risk in light of investor risk preferences

(i.e., risk seeking, risk aversion). Summarize your findings in a one-page white paper.

White papers are reports for purposes such as educating customers or demonstrating successful applications of a companys products. White papers should be free of jargon, short in length (a page or two), and easily understood by the average consumer.

Summarize your findings

When there is a difference in risk measures and expected return it denotes that risk aversion investors are usually contented with a lesser risk portfolio regardless of its lower expected returns.

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