186 Chapter 4 Discrete Probability Distributions
4.7 Jointly Distributed Discrete Random Variables 187 Portfolio analysis developed using discrete random variables is expanded in Chapter 5 using continuous random variables. The development here using discrete random vari-ables is more intuitive compared to using continuous random varivari-ables. However, the results for means, variances, covariances, and linear combinations of random variables also apply directly to continuous random variables. Since portfolios involve prices that are continuous random variables, the development in Chapter 5 is more realistic. In ad-dition, the normal distribution developed in Chapter 5 provides important analysis tools.
The variance for W is s2
W = a2s2
X + b2s2
Y + 2abCov1X, Y2 (4.43)
or, using the correlation, is s2
W = a2s2
X + b2s2
Y + 2abCorr1X, Y2sXsY
Example 4.19 Analysis of Stock Portfolios (Means and Variances, Functions of Random Variables)
George Tiao has 5 shares of stock A and 10 shares of stock B, whose price variations are modeled by the probability distribution in Table 4.8. Find the mean and variance of the portfolio.
Table 4.8 Joint Probability Distribution for Stock A and Stock B Prices STOCK A
PRICE
STOCK B PRICE
$40 $50 $60 $70
$45 0.24 0.003333 0.003333 0.003333
$50 0.003333 0.24 0.003333 0.003333
$55 0.003333 0.003333 0.24 0.003333
$60 0.003333 0.003333 0.003333 0.24
Solution The value, W, of the portfolio can be represented by the linear combination W = 5X + 10Y
Using the probability distribution in Table 4.8 we can compute the means, variances, and covariances for the two stock prices. The mean and variance for stock A are $53 and 31.3, respectively, while for stock B they are $55 and 125. The covariance is 59.17 and the correlation is 0.947.
The mean value for the portfolio is as follows:
mW = E3W4 = E35X + 10Y4 = 51532 + 11021552 = $815 The variance for the portfolio value is as follows:
s2W = 52s2X + 102s2Y + 2 * 5 * 10 * Cov1X, Y2
= 52 * 31.3 + 102 * 125 + 2 * 5 * 10 * 59.17 = 19,199.5
George knows that high variance implies high risk. He believes that the risk for this portfolio is too high. Thus, he asks you to prepare a portfolio that has lower risk. After some investigation you discover a different pair of stocks whose prices follow the prob-ability distribution in Table 4.9. By comparing Tables 4.8 and 4.9 we note that the stock prices tend to change directly with each other in Table 4.8, while they move in opposite directions in Table 4.9.
188 Chapter 4 Discrete Probability Distributions
Table 4.9 Probability Distribution for New Portfolio of Stock C and Stock D STOCK C
PRICE
STOCK D PRICE
$40 $50 $60 $70
$45 0.003333 0.003333 0.003333 0.24
$50 0.003333 0.003333 0.24 0.003333
$55 0.003333 0.24 0.003333 0.003333
$60 0.24 0.003333 0.003333 0.003333
Using the probability distribution in Table 4.9 we computed the means, variances, and covariance for the new stock portfolio. The mean for stock C is $53, the same as for stock A. Similarly, the mean for stock D is $55, the same as for stock B. Thus, the mean value of the portfolio is not changed. The variance for each stock is also the same, but the covariance is now –59.17. Thus, the variance for the new portfolio includes a nega-tive covariance term and is as follows:
s2W = 52s2
X + 102s2
Y + 2 * 5 * 10 * Cov1X, Y2
= 52 * 31.3 + 102 * 125 + 2 * 5 * 10 * 1 -59.172 = 7,365.5
We see that the effect of the negative covariance is to reduce the variance and, hence, to reduce the risk of the portfolio.
Figure 4.6 shows how portfolio variance—and, hence, risk—changes with different correlations between stock prices. Note that the portfolio variance is linearly related to the correlation. To help control risk, designers of stock portfolios select stocks based on the correlation between prices.
Figure 4.6 Portfolio Variance Versus Correlation of Stock Prices
As we saw in Example 4.19, the correlation between stock prices, or between any two random variables, has important effects on the portfolio value random variable. A posi-tive correlation indicates that both prices, X and Y, increase or decrease together. Thus, large or small values of the portfolio are magnified, resulting in greater range and vari-ance compared to a zero correlation. Conversely, a negative correlation leads to price in-creases for X matched by price dein-creases for Y. As a result, the range and variance of the portfolio are decreased compared to a zero correlation. By selecting stocks with particu-lar combinations of correlations, fund managers can control the variance and the risk for portfolios.
Exercises 189 Basic Exercises
4.71 A call center in Perth, Australia receives an average of 1.3 calls per minute. By looking at the date, a Poisson discrete distribution is assumed for this variable. Cal-culate each of the following.
a. The probability of receiving no calls in the first minute of its office hours.
b. The probability of receiving 1 call in the first minute.
c. The probability of receiving 3 calls in the first minute.
4.72 Consider the joint probability distribution:
X
1 2
Y 0 0.25 0.25
1 0.25 0.25
a. Compute the marginal probability distributions for X and Y.
b. Compute the covariance and correlation for X and Y.
c. Compute the mean and variance for the linear function W = X + Y.
4.73 Consider the joint probability distribution:
X
1 2
Y 0 0.30 0.20
1 0.25 0.25
a. Compute the marginal probability distributions for X and Y.
b. Compute the covariance and correlation for X and Y.
c. Compute the mean and variance for the linear function W = 2X + Y.
4.74 Consider the joint probability distribution:
X
1 2
Y 0 0.70 0.0
1 0.0 0.30
a. Compute the marginal probability distributions for X and Y.
b. Compute the covariance and correlation for X and Y.
c. Compute the mean and variance for the linear function W = 3X + 4Y.
4.75 Consider the joint probability distribution:
X
1 2
Y 0 0.0 0.60
1 0.40 0.0
a. Compute the marginal probability distributions for X and Y.
b. Compute the covariance and correlation for X and Y.
c. Compute the mean and variance for the linear function W = 2X - 4Y.
4.76 Consider the joint probability distribution:
X
1 2
Y 0 0.70 0.0
1 0.0 0.30
a. Compute the marginal probability distributions for X and Y.
b. Compute the covariance and correlation for X and Y.
c. Compute the mean and variance for the linear function W = 10X - 8Y.
Application Exercises
4.77 A researcher suspected that the number of between-meal snacks eaten by students in a day during final examinations might depend on the number of tests a student had to take on that day. The accompany-ing table shows joint probabilities, estimated from a survey.
Number of Snacks (Y)
Number of Tests (X)
0 1 2 3
0 0.07 0.09 0.06 0.01
1 0.07 0.06 0.07 0.01
2 0.06 0.07 0.14 0.03
3 0.02 0.04 0.16 0.04
a. Find the probability distribution of X and compute the mean number of tests taken by students on that day.
b. Find the probability distribution of Y and, hence, the mean number of snacks eaten by students on that day.
c. Find and interpret the conditional probability dis-tribution of Y, given that X = 3.
d. Find the covariance between X and Y.
e. Are number of snacks and number of tests indepen-dent of each other?
4.78 A real estate agent is interested in the relationship be-tween the number of lines in a newspaper advertise-ment for an apartadvertise-ment and the volume of inquiries from potential renters. Let volume of inquiries be de-noted by the random variable X, with the value 0 for little interest, 1 for moderate interest, and 2 for strong interest. The real estate agent used historical records to compute the joint probability distribution shown in the accompanying table.
Number of Lines (Y)
Number of Inquiries (X)
0 1 2
3 0.09 0.14 0.07
4 0.07 0.23 0.16
5 0.03 0.10 0.11
E
XERCISES190 Chapter 4 Discrete Probability Distributions a. Find the joint cumulative probability at
X= 1, Y = 4, and interpret your result.
b. Find and interpret the conditional probability distribution for Y, given X = 0.
c. Find and interpret the conditional probability distribution for X, given Y = 4.
d. Find and interpret the covariance between X and Y.
e. Are number of lines in the advertisement and volume of inquiries independent of one another?
4.79 The accompanying table shows, for credit-card hold-ers with one to three cards, the joint probabilities for number of cards owned (X) and number of credit pur-chases made in a week (Y).
Number of Cards (X)
Number of Purchases in Week (Y)
0 1 2 3 4
1 0.08 0.13 0.09 0.06 0.03
2 0.03 0.08 0.08 0.09 0.07
3 0.01 0.03 0.06 0.08 0.08
a. For a randomly chosen person from this group, what is the probability distribution for number of purchases made in a week?
b. For a person in this group who has three cards, what is the probability distribution for number of purchases made in a week?
c. Are number of cards owned and number of purchases made statistically independent?
4.80 A market researcher wants to determine whether a new model of a personal computer that had been ad-vertised on a late-night talk show had achieved more brand-name recognition among people who watched the show regularly than among people who did not.
After conducting a survey, it was found that 15% of all people both watched the show regularly and could correctly identify the product. Also, 16% of all people regularly watched the show and 45% of all people could correctly identify the product. Define a pair of random variables as follows:
X= 1 if regularly watch the show X= 0 otherwise Y= 1 if product correctly identified Y= 0 otherwise a. Find the joint probability distribution of X and Y.
b. Find the conditional probability distribution of Y, given X = 1.
c. Find and interpret the covariance between X and Y.
4.81 A college bookseller makes calls at the offices of profes-sors and forms the impression that profesprofes-sors are more likely to be away from their offices on Friday than any other working day. A review of the records of calls, 1/5 of which are on Fridays, indicates that for 16%
of Friday calls, the professor is away from the office, while this occurs for only 12% of calls on every other working day. Define the random variables as follows:
X= 1 if call is made on a Friday X= 0 otherwise Y= 1 if professor is away from the office Y= 0 otherwise
a. Find the joint probability distribution of X and Y.
b. Find the conditional probability distribution of Y, given X = 0.
c. Find the marginal probability distributions of X and Y.
d. Find and interpret the covariance between X and Y.
4.82 A restaurant manager receives occasional complaints about the quality of both the food and the service. The marginal probability distributions for the number of weekly complaints in each category are shown in the accompanying table. If complaints about food and ser-vice are independent of each other, find the joint prob-ability distribution.
Number of Food
Complaints Probability
Number of Service
Complaints Probability
0 0.12 0 0.18
1 0.29 1 0.38
2 0.42 2 0.34
3 0.17 3 0.10
4.83 Refer to the information in the previous exercise.
Find the mean and standard deviation of the to-tal number of complaints received in a week. Hav-ing reached this point, you are concerned that the numbers of food and service complaints may not be independent of each other. However, you have no information about the nature of their dependence.
What can you now say about the mean and standard deviation of the total number of complaints received in a week?
4.84 A company has 5 representatives covering large ritories and 10 representatives covering smaller ter-ritories. The probability distributions for the numbers of orders received by each of these types of represen-tatives in a day are shown in the accompanying table.
Assuming that the number of orders received by any representative is independent of the number received by any other, find the mean and standard deviation of the total number of orders received by the com-pany in a day.
Numbers of Orders (Large
Territories) Probability
Numbers of Orders (Smaller
Territories) Probability
0 0.08 0 0.18
1 0.16 1 0.26
2 0.28 2 0.36
3 0.32 3 0.13
4 0.10 4 0.07
5 0.06
Chapter Exercises and Applications 191