Uncorrelatedness (probability theory)

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In probability theory and statistics, two real-valued random variables, ${\displaystyle X}$, ${\displaystyle Y}$, are said to be uncorrelated if their covariance, ${\displaystyle \operatorname {cov} [X,Y]=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]}$, 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 ${\displaystyle X}$ and ${\displaystyle Y}$ are uncorrelated if and only if ${\displaystyle \operatorname {E} [XY]=0}$.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are independent, with finite second moments, then they are uncorrelated. However, not all uncorrelated variables are independent.^{[1]: p. 155}