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 μy, variances σ2x and σ2y, and population correlation coefficient ρ (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula. μw = aμx + bμyσ2w = a2σ2x + b2σ2y + 2abσxσyρ In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σxσyρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ2w 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. μx ≈ 7.31, σx ≈ 6.55, μy ≈ 13.17, σy ≈ 18.57, ρ ≈ 0.423(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 σw.μw = σw = (c) Repeat part (b) if w = 0.3x + 0.7y.μw = σw =

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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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 μ_x and μ_y, variances σ²_x and σ²_y, and population correlation coefficient ρ (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula.

μ_w = aμ_x + bμ_y
σ²_w = a²σ²_x + b²σ²_y + 2abσ_xσ_yρ

In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σ_xσ_yρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ²_w 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.

μ_x ≈ 7.31, σ_x ≈ 6.55, μ_y ≈ 13.17, σ_y ≈ 18.57, ρ ≈ 0.423(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 μ_w and risk σ_w.
μ_w =
σ_w =

(c) Repeat part (b) if w = 0.3x + 0.7y.
μ_w =
σ_w =

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