In probability theory and statistics, two real-valued random variables are said to be uncorrelated if their covariance is zero.

Uncorrelated random variables have a correlation coefficient of zero, except in the trivial case when both variables have variance zero (are constants). 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) = E("X")E("Y").

If "X" and "Y" are independent, 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" = "X"2, 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".

Uncorrelatedness is a relation between only two random variables. By contrast, independence can be a relationship between more than two.

ee also

*Bivariate normal distribution#Correlations_and_independence

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