Uncorrelatedness (probability theory)

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In probability theory and statistics, two real-valued random variables, , , are said to be uncorrelated if their covariance, , is zero. 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 at least one of the two random variables has an expected value of 0. In this case, the covariance is the expectation of the product, and and are uncorrelated if and only if .

If and are independent, with finite second moments, then they are uncorrelated. However, not all uncorrelated variables are independent.[1]:p. 155

Definition[edit]

Definition for two real random variables[edit]

Two random variables are called uncorrelated if their covariance is zero[1]:p. 153[2]:p. 121. Formally:

Definition for two complex random variables[edit]

Two complex random variables are called uncorrelated if their covariance and their pseudo-covariance is zero, i.e.

Definition for more than two random variables[edit]

A set of two or more random variables is called uncorrelated if each pair of them is uncorrelated. This is equivalent to the requirement that the non-diagonal elements of the autocovariance matrix of the random vector are all zero. The autocovariance matrix is defined as:

Examples of dependence without correlation[edit]

Example 1[edit]

  • Let be a random variable that takes the value 0 with probability 1/2, and takes the value 1 with probability 1/2.
  • Let be a random variable, independent of , that takes the value −1 with probability 1/2, and takes the value 1 with probability 1/2.
  • Let be a random variable constructed as .

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

Proof:

First note:

where the second equality holds because and are independent. Now,

So and are uncorrelated.

Independence of and means that for all and , . This is not true, in particular, for and .

Thus so and are not independent.

Q.E.D.

Example 2[edit]

If is a continuous random variable uniformly distributed on and , then and are uncorrelated even though determines and a particular value of can be produced by only one or two values of .

When uncorrelatedness implies independence[edit]

There 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 Bernoulli distribution).[3] Further, two jointly normally distributed random variables are independent if they are uncorrelated,[4] 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).

Generalizations[edit]

Uncorrelated random vectors[edit]

Two random vectors and are called uncorrelated if

.

They are uncorrelated if and only if their cross-covariance matrix is zero.[5]:p.337

Two complex random vectors and are called uncorrelated if their cross-covariance matrix and their pseudo-cross-covariance matrix is zero, i.e. if

where

and

.

Uncorrelated stochastic processes[edit]

Two stochastic processes and are called uncorrelated if their cross-covariance is zero for all times.[2]:p. 142 Formally:

See also[edit]

References[edit]

  1. ^ a b Papoulis, Athanasios (1991). Probability, Random Variables and Stochastic Porcesses. MCGraw Hill. ISBN 0-07-048477-5.
  2. ^ a b Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3
  3. ^ Virtual Laboratories in Probability and Statistics: Covariance and Correlation, item 17.
  4. ^ Bain, Lee; Engelhardt, Max (1992). "Chapter 5.5 Conditional Expectation". Introduction to Probability and Mathematical Statistics (2nd ed.). pp. 185–186. ISBN 0534929303.
  5. ^ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1.

Further reading[edit]

  • Probability for Statisticians, Galen R. Shorack, Springer (c2000) ISBN 0-387-98953-6