#jsDisabledContent { display:none; } My Account | Register | Help
 Flag as Inappropriate This article will be permanently flagged as inappropriate and made unaccessible to everyone. Are you certain this article is inappropriate?          Excessive Violence          Sexual Content          Political / Social Email this Article Email Address:

# Uncorrelated random variables

Article Id: WHEBN0000173981
Reproduction Date:

 Title: Uncorrelated random variables Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Uncorrelated random variables

In probability theory and statistics, two real-valued random variables, X,Y, are said to be uncorrelated if their covariance, E(XY) − E(X)E(Y), is zero. A set of two or more random variables is called uncorrelated if each pair of them are uncorrelated. If two variables are uncorrelated, there is no linear relationship between them.

Uncorrelated random variables have a Pearson correlation coefficient of zero, except in the trivial case when either variable has zero variance (is a constant). In this case the correlation is undefined.

In general, uncorrelatedness is not the same as orthogonality, except in the special case where either X or Y has an expected value of 0. In this case, the covariance is the expectation of the product, and X and Y are uncorrelated if and only if E(XY) = 0.

If X and Y are independent, with finite second moments, then they are uncorrelated. However, not all uncorrelated variables are independent. For example, if X is a continuous random variable uniformly distributed on [−1, 1] and Y = X2, then X and Y are uncorrelated even though X determines Y and a particular value of Y can be produced by only one or two values of X.

## Contents

• Example of dependence without correlation 1
• When uncorrelatedness implies independence 2
• See also 3
• References 4
• Further reading 5

## Example of dependence without correlation

Uncorrelated random variables are not necessarily independent
• Let X be a random variable that takes the value 0 with probability 1/2, and takes the value 1 with probability 1/2.
• Let Z be a random variable, independent of X, that takes the value −1 with probability 1/2, and takes the value 1 with probability 1/2.
• Let U be a random variable constructed as U = XZ.

The claim is that U and X have zero covariance (and thus are uncorrelated), but are not independent.

Proof:

First note:

• E[U] = E[XZ] = E[X] E[Z] = E[X] \cdot 0 = 0

Now, \operatorname{cov}(U,X) = E[(U-E[U])(X-E[X])] = E[ U (X-\tfrac12)] = E[X^2Z - \tfrac12 XZ] = E[(X^2-\tfrac12 X)Z] = E[(X^2-\tfrac12 X)] E[Z] = 0

Independence of U and X means that for all a and b, \Pr(U=a\mid X=b) = \Pr(U=a). This is not true, in particular, for a = 1 and b = 0.

• \Pr(U=1\mid X=0) = \Pr(XZ=1\mid X=0) = 0
• \Pr(U=1) = \Pr(XZ=1) = 1/4

Thus \Pr(U=1\mid X=0)\ne \Pr(U=1) so U and X are not independent.

Q.E.D.

## When uncorrelatedness implies independence

Here are cases in which uncorrelatedness does imply independence. One of these cases is the one in which both random variables are two-valued (so each can be linearly transformed to have a binomial distribution with n = 1).[1] Further, two jointly normally distributed random variables are independent if they are uncorrelated,[2] although this does not hold for variables whose marginal distributions are normal and uncorrelated but whose joint distribution is not joint normal (see Normally distributed and uncorrelated does not imply independent).

## References

1. ^ Virtual Laboratories in Probability and Statistics: Covariance and Correlation, item 17.
2. ^ Bain, Lee; Engelhardt, Max (1992). "Chapter 5.5 Conditional Expectation". Introduction to Probability and Mathematical Statistics (2nd ed.). pp. 185–186.

## Further reading

• Probability for Statisticians, Galen R. Shorack, Springer (c2000) ISBN 0-387-98953-6
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.

By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.