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3. Let X₁ and X₂ be a pair of random variables. Show that the covariance between the random variables (X₁ + X₂) and (X₁ – X₂) is 0 if and only if X₁ and X₂ have the same variance.

3. Let X₁ and X₂ be a pair of random variables. Show that the covariance between the random variables (X₁ + X₂) and (X₁ – X₂) is 0 if and only if X₁ and X₂ have the same variance.

MATLAB: An Introduction with Applications
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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These questions are in most cases taken from Newbold et al. For numerical questions, obtain an answer both with a computer program such as Matlab, R, or Python and with tables if you are able to do so, but solve with tables in any event. 1. A random variable has the density function: fx(x) = 0, if x < 3; x − 3, if 3 ≤ x ≤ 4; 5 – x, if 4 < x < 5; 0, x ≥ 5. Using the expression for the expectation (mean) of a continuous random vari- able, i.e. E(X) = xfx(x)dx, treating the two intervals 3 ≤ x ≤ 4 and 4 ≤ x ≤ 5 separately, show that the mean of this random variable is 4. 2. (a) Using the expression for variance in either the discrete or continuous case, show that for a random variable X and fixed parameters a and b, var(a + bx) = b²var (X). (b) Using the expression K(X) = E(X−µ) as a kurtosis measure, show that for a random variable X and fixed parameters a and b, K(a + bX) = K(X). Interpret this with respect to the standardization by ¹. 3. Let X₁ and X₂ be a pair of random variables. Show that the covariance between the random variables (X₁ + X₂) and (X₁ – X₂) is 0 if and only if X₁ and X₂ have the same variance.
Transcribed Image Text:These questions are in most cases taken from Newbold et al. For numerical questions, obtain an answer both with a computer program such as Matlab, R, or Python and with tables if you are able to do so, but solve with tables in any event. 1. A random variable has the density function: fx(x) = 0, if x < 3; x − 3, if 3 ≤ x ≤ 4; 5 – x, if 4 < x < 5; 0, x ≥ 5. Using the expression for the expectation (mean) of a continuous random vari- able, i.e. E(X) = xfx(x)dx, treating the two intervals 3 ≤ x ≤ 4 and 4 ≤ x ≤ 5 separately, show that the mean of this random variable is 4. 2. (a) Using the expression for variance in either the discrete or continuous case, show that for a random variable X and fixed parameters a and b, var(a + bx) = b²var (X). (b) Using the expression K(X) = E(X−µ) as a kurtosis measure, show that for a random variable X and fixed parameters a and b, K(a + bX) = K(X). Interpret this with respect to the standardization by ¹. 3. Let X₁ and X₂ be a pair of random variables. Show that the covariance between the random variables (X₁ + X₂) and (X₁ – X₂) is 0 if and only if X₁ and X₂ have the same variance.
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