Previously, you studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let x and y be random variables with means µx and Hy, variances o?x and o?y, and population correlation coefficient p (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula. Hw = aµx + bµy o²w = a²o²x + b²o?y + 2abox0yp In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression o,oyp is called the covariance of x and y. If x and y are independent, then p = 0 and the formula for o²w reduces to the appropriate formula for independent variables. In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates. Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let x represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let y represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates. Hx = 7.35, Ox - 6.56, Hy = 13.19, Oy z 18.57, p z 0.422 (a) Do you think the variables x and y are independent? Explain your answer. O No. Interest rate probably affects both investment returns. O Yes. Interest rates probably has no effect on the investment returns. O No. Interest rates probably has no effect on the investment returns. O Yes. Interest rate probably affects both investment returns. (b) Suppose you decide to put 70% of your investment in bonds and 30% in real estate. This means you will use a weighted average w = 0.7x + 0.3y. Estimate your expected percentage return µw and risk ow. Hw = Ow = (c) Repeat part (b) if w = 0.3x + 0.7y. Hw = Ow =
Previously, you studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let x and y be random variables with means µx and Hy, variances o?x and o?y, and population correlation coefficient p (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula. Hw = aµx + bµy o²w = a²o²x + b²o?y + 2abox0yp In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression o,oyp is called the covariance of x and y. If x and y are independent, then p = 0 and the formula for o²w reduces to the appropriate formula for independent variables. In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates. Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let x represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let y represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates. Hx = 7.35, Ox - 6.56, Hy = 13.19, Oy z 18.57, p z 0.422 (a) Do you think the variables x and y are independent? Explain your answer. O No. Interest rate probably affects both investment returns. O Yes. Interest rates probably has no effect on the investment returns. O No. Interest rates probably has no effect on the investment returns. O Yes. Interest rate probably affects both investment returns. (b) Suppose you decide to put 70% of your investment in bonds and 30% in real estate. This means you will use a weighted average w = 0.7x + 0.3y. Estimate your expected percentage return µw and risk ow. Hw = Ow = (c) Repeat part (b) if w = 0.3x + 0.7y. Hw = Ow =
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:Previously, you studied linear combinations of independent random
variables. What happens if the variables are not independent? A lot of
mathematics can be used to prove the following: Let x and y be random
variables with means µx and Hy, variances o?x and o?y, and population
correlation coefficient p (the Greek letter rho). Let a and b be any constants
and let w = ax + by for the following formula.
Hw = aµx + bµy
o²w = a²o²x + b²o?y + 2abox0yp
In this formula, r is the population correlation coefficient, theoretically
computed using the population of all (x, y) data pairs. The expression o,oyp
is called the covariance of x and y. If x and y are independent, then p = 0
and the formula for o²w reduces to the appropriate formula for independent
variables. In most real-world applications the population parameters are not
known, so we use sample estimates with the understanding that our
conclusions are also estimates.
Do you have to be rich to invest in bonds and real estate? No, mutual fund
shares are available to you even if you aren't rich. Let x represent annual
percentage return (after expenses) on the Vanguard Total Bond Index Fund,
and let y represent annual percentage return on the Fidelity Real Estate
Investment Fund. Over a long period of time, we have the following
population estimates.
Hx = 7.35,
Ox - 6.56,
Hy = 13.19,
Oy z 18.57,
p z 0.422
(a) Do you think the variables x and y are independent? Explain your
answer.
O No. Interest rate probably affects both investment returns.
O Yes. Interest rates probably has no effect on the investment
returns.
O No. Interest rates probably has no effect on the investment returns.
O Yes. Interest rate probably affects both investment returns.
(b) Suppose you decide to put 70% of your investment in bonds and
30% in real estate. This means you will use a weighted average w =
0.7x + 0.3y. Estimate your expected percentage return µw and risk ow.
Hw =
Ow =
(c) Repeat part (b) if w = 0.3x + 0.7y.
Hw =
Ow =
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