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3. Recall that (Corollary 3.8 in the textbook) if two random variables X and Y are inde- pendent, then they must be uncorrelated i.e. Cov(X,Y)= 0. However, the converse is not true in general and this problem provides an example. Let X be a random variable with continuous uniform distribution on the interval [-1,1], i.e. its probability density function is given by f(x) = [1/2, if x € [-1,1], otherwise. a) Show that Cov(X, X²) = 0. b) Prove mathematically (not just argue by intuition) that X and X² are not inde- pendent. One way to do this is by showing that they do not satisfy the property: P(X € A, X² € B) = P(X E A) P(X² = B) for all A, BCR. You may also use other equivalent definitions of independence.

3. Recall that (Corollary 3.8 in the textbook) if two random variables X and Y are inde- pendent, then they must be uncorrelated i.e. Cov(X,Y)= 0. However, the converse is not true in general and this problem provides an example. Let X be a random variable with continuous uniform distribution on the interval [-1,1], i.e. its probability density function is given by f(x) = [1/2, if x € [-1,1], otherwise. a) Show that Cov(X, X²) = 0. b) Prove mathematically (not just argue by intuition) that X and X² are not inde- pendent. One way to do this is by showing that they do not satisfy the property: P(X € A, X² € B) = P(X E A) P(X² = B) for all A, BCR. You may also use other equivalent definitions of independence.

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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3. Recall that (Corollary 3.8 in the textbook) if two random variables X and Y are inde- pendent, then they must be uncorrelated i.e. Cov(X,Y)= 0. However, the converse is not true in general and this problem provides an example. Let X be a random variable with continuous uniform distribution on the interval [-1,1], i.e. its probability density function is given by f(x) = (1/2, if xe [-1, 1], 0, otherwise. a) Show that Cov(X, X²) = 0. b) Prove mathematically (not just argue by intuition) that X and X² are not inde- pendent. One way to do this is by showing that they do not satisfy the property: P(X E A, X² € B) = P(X € A) · P(X² € B) . for all A, BCR. You may also use other equivalent definitions of independence.
Transcribed Image Text:3. Recall that (Corollary 3.8 in the textbook) if two random variables X and Y are inde- pendent, then they must be uncorrelated i.e. Cov(X,Y)= 0. However, the converse is not true in general and this problem provides an example. Let X be a random variable with continuous uniform distribution on the interval [-1,1], i.e. its probability density function is given by f(x) = (1/2, if xe [-1, 1], 0, otherwise. a) Show that Cov(X, X²) = 0. b) Prove mathematically (not just argue by intuition) that X and X² are not inde- pendent. One way to do this is by showing that they do not satisfy the property: P(X E A, X² € B) = P(X € A) · P(X² € B) . for all A, BCR. You may also use other equivalent definitions of independence.
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