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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, when it exists, 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.

Two random variables are called uncorrelated if their covariance is zero. Formally:

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

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:

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). Further, two jointly normally distributed random variables are independent if they are uncorrelated, 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).

Two stochastic processes and are called uncorrelated if their cross covariance is zero for all times. Formally:

.