Homework Help is Here – Start Your Trial Now!
Math
Statistics
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
See similar textbooks
icon
Related questions
Question
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.
Expert Solution
steps

Step by step

Solved in 4 steps with 24 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman
SEE MORE TEXTBOOKS