Definition for two real random variables

Two random variables ${\displaystyle X,Y}$ are called uncorrelated if their covariance ${\displaystyle \operatorname {Cov} [X,Y]=\operatorname {E} [(X-\operatorname {E} [X])(Y-\operatorname {E} [Y])]}$ is zero.^{[1]: p. 153 [2]: p. 121} Formally:

Definition for two complex random variables

Two complex random variables ${\displaystyle Z,W}$ are called uncorrelated if their covariance ${\displaystyle \operatorname {K} _{ZW}=\operatorname {E} [(Z-\operatorname {E} [Z]){\overline {(W-\operatorname {E} [W])}}]}$ and their pseudo-covariance ${\displaystyle \operatorname {J} _{ZW}=\operatorname {E} [(Z-\operatorname {E} [Z])(W-\operatorname {E} [W])]}$ is zero, i.e.

${\displaystyle Z,W{\text{ uncorrelated}}\quad \iff \quad \operatorname {E} [Z{\overline {W}}]=\operatorname {E} [Z]\cdot \operatorname {E} [{\overline {W}}]{\text{ and }}\operatorname {E} [ZW]=\operatorname {E} [Z]\cdot \operatorname {E} [W]}$

Definition for more than two random variables

A set of two or more random variables ${\displaystyle X_{1},\ldots ,X_{n}}$ 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 ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}$ of the random vector ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathrm {T} }}$ are all zero. The autocovariance matrix is defined as:

${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {cov} [\mathbf {X} ,\mathbf {X} ]=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\rm {T}}]=\operatorname {E} [\mathbf {X} \mathbf {X} ^{T}]-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{T}}$

Example 1

• Let ${\displaystyle X}$ be a random variable that takes the value 0 with probability 1/2, and takes the value 1 with probability 1/2.
• Let ${\displaystyle Y}$ be a random variable, independent of ${\displaystyle X}$, that takes the value −1 with probability 1/2, and takes the value 1 with probability 1/2.
• Let ${\displaystyle U}$ be a random variable constructed as ${\displaystyle U=XY}$.

The claim is that ${\displaystyle U}$ and ${\displaystyle X}$ have zero covariance (and thus are uncorrelated), but are not independent.

Proof:

Taking into account that

${\displaystyle \operatorname {E} [U]=\operatorname {E} [XY]=\operatorname {E} [X]\operatorname {E} [Y]=\operatorname {E} [X]\cdot 0=0,}$

where the second equality holds because ${\displaystyle X}$ and ${\displaystyle Y}$ are independent, one gets

${\displaystyle {\begin{aligned}\operatorname {cov} [U,X]&=\operatorname {E} [(U-\operatorname {E} [U])(X-\operatorname {E} [X])]=\operatorname {E} [U(X-{\tfrac {1}{2}})]\\&=\operatorname {E} [X^{2}Y-{\tfrac {1}{2}}XY]=\operatorname {E} [(X^{2}-{\tfrac {1}{2}}X)Y]=\operatorname {E} [(X^{2}-{\tfrac {1}{2}}X)]\operatorname {E} [Y]=0\end{aligned}}}$

Therefore, ${\displaystyle U}$ and ${\displaystyle X}$ are uncorrelated.

Independence of ${\displaystyle U}$ and ${\displaystyle X}$ means that for all ${\displaystyle a}$ and ${\displaystyle b}$, ${\displaystyle \Pr(U=a\mid X=b)=\Pr(U=a)}$. This is not true, in particular, for ${\displaystyle a=1}$ and ${\displaystyle b=0}$.

• ${\displaystyle \Pr(U=1\mid X=0)=\Pr(XY=1\mid X=0)=0}$
• ${\displaystyle \Pr(U=1)=\Pr(XY=1)=1/4}$

Thus ${\displaystyle \Pr(U=1\mid X=0)\neq \Pr(U=1)}$ so ${\displaystyle U}$ and ${\displaystyle X}$ are not independent.

Q.E.D.

Example 2

If ${\displaystyle X}$ is a continuous random variable uniformly distributed on ${\displaystyle [-1,1]}$ and ${\displaystyle Y=X^{2}}$, then ${\displaystyle X}$ and ${\displaystyle Y}$ are uncorrelated even though ${\displaystyle X}$ determines ${\displaystyle Y}$ and a particular value of ${\displaystyle Y}$ can be produced by only one or two values of ${\displaystyle X}$ :

${\displaystyle f_{X}(t)={1 \over 2}I_{[-1,1]};f_{Y}(t)={1 \over {2{\sqrt {t}}}}I_{]0,1]}}$

on the other hand, ${\displaystyle f_{X,Y}}$ is 0 on the triangle defined by ${\displaystyle 0<X<Y<1}$ although ${\displaystyle f_{X}\times f_{Y}}$ is not null on this domain. Therefore ${\displaystyle f_{X,Y}(X,Y)\neq f_{X}(X)\times f_{Y}(Y)}$ and the variables are not independent.

${\displaystyle E[X]={{1-1} \over 4}=0;E[Y]={{1^{3}-(-1)^{3}} \over {3\times 2}}={1 \over 3}}$

${\displaystyle Cov[X,Y]=E\left[(X-E[X])(Y-E[Y])\right]=E\left[X^{3}-{X \over 3}\right]={{1^{4}-(-1)^{4}} \over {4\times 2}}=0}$

Therefore the variables are uncorrelated.

Uncorrelated random vectors

Two random vectors ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{T}}$ and ${\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{T}}$ are called uncorrelated if

${\displaystyle \operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}$.

They are uncorrelated if and only if their cross-covariance matrix ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}$ is zero.^[5]: p.337

Two complex random vectors ${\displaystyle \mathbf {Z} }$ and ${\displaystyle \mathbf {W} }$ are called uncorrelated if their cross-covariance matrix and their pseudo-cross-covariance matrix is zero, i.e. if

${\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {J} _{\mathbf {Z} \mathbf {W} }=0}$

where

${\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{\mathrm {H} }]}$

and

${\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {W} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{\mathrm {T} }]}$.

Uncorrelated stochastic processes

Two stochastic processes ${\displaystyle \left\{X_{t}\right\}}$ and ${\displaystyle \left\{Y_{t}\right\}}$ are called uncorrelated if their cross-covariance ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} \left[\left(X(t_{1})-\mu _{X}(t_{1})\right)\left(Y(t_{2})-\mu _{Y}(t_{2})\right)\right]}$ is zero for all times.^{[2]: p. 142} Formally:

${\displaystyle \left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ uncorrelated}}\quad$ :\iff \quad \forall t_{1},t_{2}\colon \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0} .