*For Exercise 4.109, see page 220; for 4.110, see page 223; for 4.111 to 4.114, see page 228; for 4.115, see page 230; and for 4.116 to 4.118, see page 235.*

4.119 Portfolio analysis.

**CASE 4.3** Show that if 20% of the portfolio is based on the S&P 500 index, then the mean and standard deviation of the portfolio are indeed the values given in Example 4.41 (page 234).

4.120 Find some means.

Suppose that is a random variable with mean 20 and standard deviation 5. Also suppose that is a random variable with mean 40 and standard deviation 10. Find the mean of the random variable for each of the following cases. Be sure to show your work.

- .
- .
- .
- .
- .

4.121 Find the variance and the standard deviation.

A random variable has the following distribution.

−1 | 0 | 1 | 2 | |

Probability | 0.3 | 0.2 | 0.2 | 0.3 |

Find the variance and the standard deviation for this random variable. Show your work.

4.122 Find some variances and standard deviations.

Suppose that is a random variable with mean 20 and standard deviation 5. Also suppose that is a random variable with mean 40 and standard deviation 10. Assume that and are independent. Find the variance and the standard deviation of the random variable for each of the following cases. Be sure to show your work.

- .
- .
- .
- .
- .

4.123 What happens if the correlation is not zero?

Suppose that is a random variable with mean 20 and standard deviation 5. Also suppose that is a random variable with mean 40 and standard deviation 10. Assume that the correlation between and is 0.5. Find the variance and standard deviation of the random variable for each of the following cases. Be sure to show your work.

- .
- .
- .

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4.124 What's wrong?

In each of the following scenarios, there is something wrong. Describe what is wrong, and give a reason for your answer.

- If you toss a fair coin three times and get heads all three times, then the probability of getting a tail on the next toss is much greater than one-half.
- If you multiply a random variable by 10, then the mean is multiplied by 10 and the variance is multiplied by 10.
- When finding the mean of the sum of two random variables, you need to know the correlation between them.

4.125 Difference between heads and tails.

Suppose a fair coin is tossed three times.

- Using the labels of “H” and “T,” list all the possible outcomes in the sample space.
- For each outcome in the sample space, define the random variable as the number of heads minus the number of tails observed. Use the fact that all outcomes of part (a) are equally likely to find the probability distribution of .
- Use the probability distribution found in (b) to find t mean and standard deviation of .

4.126 Mean of the distribution for the number of aces.

In Exercise 4.98 (page 217), you examined the probability distribution for the number of aces when you are dealt two cards in the game of Texas hold 'em. Let represent the number of aces in a randomly selected deal of two cards in this game. Here is the probability distribution for the random variable :

Value of | 0 | 1 | 2 |

Probability | 0.8507 | 0.1448 | 0.0045 |

Find , the mean of the probability distribution of .

4.127 Standard deviation of the number of aces.

Refer to the previous exercise. Find the standard deviation of the number of aces.

4.128 Difference between heads and tails.

In Exercise 4.125, the mean and standard deviation were computed directly from the probability distribution of random variable . Instead, define as the number of heads in the three flips, and define as the number of tails in the three flips.

- Find the probability distribution for along with the mean and standard deviation .
- Find the probability distribution for along with the mean and standard deviation .
- Explain why the correlation between and is .
- Define as . Use the rules of means and variances along with to find the mean and standard deviation of . Confirm the values are the same as found in Exercise 4.125.

4.129 Pick 3 and law of large numbers.

In Example 4.28 (pages 219–220), the mean payoff for the Tri-State Pick 3 lottery was found to be $0.50. In our discussion of the law of large numbers, we learned that the mean of a probability distribution describes the long-run average outcome. In this exercise, you will explore this concept using technology.

*Excel users:*Input the values “0” and “500” in the first two rows of column A. Now input the corresponding probabilities of 0.999 and 0.001 in the first two rows of column B. Now choose “Random Number Generation” from the**Data Analysis**menu box. Enter “1” in the**Number of Variables**box, enter “20000” in the**Number of Random Numbers**box, choose “Discrete” for the**Distribution**option, enter the cell range of the -values and their probabilities ($A$1:$B$2) in**Value and Probability Input Range**box, and finally select Row 1 of any empty column for the**Output Range**. Click**OK**to find 20,000 realizations of outputted in the worksheet. Using Excel's AVERAGE() function, find the average of the 20,000 -values.*JMP users:*With a new data table, right-click on header of Column 1 and choose**Column Info**. In the drag-down dialog box named**Initialize Data**, pick**Random**option. Choose the bullet option of**Random Indicator**. Put the values of “0” and “500” in the first two**Value**dialog boxes, and put the values of 0.999 and 0.001 in the corresponding**Proportion**dialog boxes. Input the Enter “20000” into the**Number of rows**box, and then click**OK**. Find the average of the 20,000 -values.*Minitab users:*Input the values “0” and “500” in the first two rows of column 1 (c1). Now input the corresponding probabilities of 0.999 and 0.001 in the first two rows of column 2 (c2). Do the following pull-down sequence: Calc→ Random Data→ Discrete. Enter “20000” in the**Number of rows of data to generate**box, type “c3” in the**Store in column(s)**box, click-in “c1” in the**Values in**box, and click-in “c2” in the**Probabilities in**box. Click**OK**to find 20,000 realizations of outputted in the worksheet. Find the average of the 20,000 -values.

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Whether you used Excel, JMP, or Minitab, how does the average value of the 20,000 -values compare with the mean reported in Example 4.28?

4.130 Households and families in government data.

In government data, a household consists of all occupants of a dwelling unit, while a family consists of two or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and of family size in the United States:

Number of persons |
1 | 2 | 3 | 4 | 5 | 6 | 7 |

Household probability |
0.27 | 0.33 | 0.16 | 0.14 | 0.06 | 0.03 | 0.01 |

Family probability |
0.00 | 0.44 | 0.22 | 0.20 | 0.09 | 0.03 | 0.02 |

Compare the two distributions using probability histograms on the same scale. Also compare the two distributions using means and standard deviations. Write a summary of your comparisons using your calculations to back up your statements.

4.131 Perfectly negatively correlated investments.

**CASE 4.3** Consider the following quote from an online site providing investment guidance: “Perfectly negatively correlated investments would provide 100% diversification, as they would form a portfolio with zero variance, which translates to zero risk.” Consider a portfolio based on two investments ( and ) with standard deviations of and . In line with the quote, assume that the two investments are perfectly negatively correlated .

- Suppose , and the portfolio mix is 70/30 of X to Y. What is the standard deviation of the portfolio? Does the portfolio have zero risk?
- Suppose , and the portfolio mix is 50/50. What is the standard deviation of the portfolio? Does the portfolio have zero risk?
- Suppose , and the portfolio mix is 50/50. What is the standard deviation of the portfolio? Does the portfolio have zero risk?
- Is the online quote a universally true statement? If not, how would you modify it so that it can be stated that the portfolio has zero risk?

4.132 What happens when the correlation is 1?

We know that variances add if the random variables involved are uncorrelated , but not otherwise. The opposite extreme is perfect positive correlation . Show by using the general addition rule for variances that in this case the standard deviations add. That is, if .

4.133 Making glassware.

In a process for manufacturing glassware, glass stems are sealed by heating them in a flame. The temperature of the flame varies. Here is the distribution of the temperature measured in degrees Celsius:

Temperature | 540° | 545° | 550° | 555° | 560° |

Probability | 0.1 | 0.25 | 0.3 | 0.25 | 0.1 |

- Find the mean temperature and the standard deviation .
- The target temperature is 550
^{o}C. Use the rules for means and variances to find the mean and standard deviation of the number of degrees off target, . - A manager asks for results in degrees Fahrenheit. The conversion of into degrees Fahrenheit is given by

What are the mean and standard deviation of the temperature of the fame in the Fahrenheit scale?

Portfolio analysis.

**CASE 4.3** *Here are the means, standard deviations, and correlations for the monthly returns from three Fidelity mutual funds for the 60 months ending in July 2014. Because there are three random variables, there are three correlations. We use subscripts to show which pair of random variables a correlation refers to**.*

*Correlations*

*Exercises 4.134 through 4.136 make use of these historical data.*

4.134 Diversification.

**CASE 4.3** Currently, Michael is exclusively invested in the Fidelity Biotechnology fund. Even though the mean return for this biotechnology fund is quite high, it comes with greater volatility and risk. So, he decides to diversify his portfolio by constructing a portfolio of 80% biotechnology fund and 20% information services fund. Based on the provided historical performance, what is the expected return and standard deviation of the portfolio? Relative to his original investment scheme, what is the percentage reduction in his risk level (as measured by standard deviation) by going to this particular portfolio?

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4.135 More on diversification.

**CASE 4.3** Continuing with the previous exercise, suppose Michael's primary goal is to seek a portfolio mix of the biotechnology and information services funds that will give him *minimal* risk as measured by standard deviation of the portfolio. Compute the standard deviations for portfolios based on the proportion of biotechnology fund in the portfolio ranging from 0 to 1 in increments of 0.1. You may wish to do these calculations in Excel. What is your recommended mix of biotechnology and information services funds for Michael? What is the standard deviation for your recommended portfolio?

4.136 Larger portfolios.

**CASE 4.3** Portfolios often contain more than two investments. The rules for means and variances continue to apply, though the arithmetic gets messier. A portfolio containing proportions of Biotechnology Fund, of Information Services Fund, and of Defense and Aerospace Fund has return . Because , , and are the proportions invested in the three funds, . The mean and variance of the portfolio return are

Having seen the advantages of diversification, Michael decides to invest his funds 20% in biotechnology, 35% in information services, and 45% in defense and aerospace. What are the (historical) mean and standard deviation of the monthly returns for this portfolio?