In probability theory and statistics, two real-valued random variables, X {\displaystyle X} , Y {\displaystyle Y} , are said to be uncorrelated if their covariance, cov [ X , Y ] = E [ X Y ] − E [ X ] E [ Y ] {\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 X {\displaystyle X} and Y {\displaystyle Y} are uncorrelated if and only if E [ X Y ] = 0 {\displaystyle \operatorname {E} [XY]=0} . If X {\displaystyle X} and Y {\displaystyle Y} are independent, with finite second moments, then they are uncorrelated. However, not all uncorrelated variables are independent.
Definition
Definition for two real random variables Two random variables X , Y {\displaystyle X,Y} are called uncorrelated if their covariance Cov [ X , Y ] = E [ ( X − E [ X ] ) ( Y − E [ Y ] ) ] {\displaystyle \operatorname {Cov} [X,Y]=\operatorname {E} [(X-\operatorname {E} [X])(Y-\operatorname {E} [Y])]} is zero. Formally:
Definition for two complex random variables Two complex random variables Z , W {\displaystyle Z,W} are called uncorrelated if their covariance K Z W = E [ ( Z − E [ Z ] ) ( W − E [ W ] ) ¯ ] {\displaystyle \operatorname {K} _{ZW}=\operatorname {E} [(Z-\operatorname {E} [Z]){\overline {(W-\operatorname {E} [W])}}]} and their pseudo-covariance J Z W = E [ ( Z − E [ Z ] ) ( W − E [ W ] ) ] {\displaystyle \operatorname {J} _{ZW}=\operatorname {E} [(Z-\operatorname {E} [Z])(W-\operatorname {E} [W])]} is zero, i.e.
Z , W uncorrelated ⟺ E [ Z W ¯ ] = E [ Z ] ⋅ E [ W ¯ ] and E [ Z W ] = E [ Z ] ⋅ E [ W ] {\displaystyle Z,W{\text{ uncorrelated}}\quad \iff \quad \operatorname {E} [Z{\overline {W}}]=\operatorname {E} [Z]\cdot \operatorname {E} [{\overline {W}}]{\text{ and }}\operatorname {E} [ZW]=\operatorname {E} [Z]\cdot \operatorname {E} [W]}
Definition for more than two random variables A set of two or more random variables X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} 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 K X X {\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }} of the random vector X = [ X 1 … X n ] T {\displaystyle \mathbf {X} =[X_{1}\ldots X_{n}]^{\mathrm {T} }} are all zero. The autocovariance matrix is defined as:
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