Concept in probability theory
In probability theory and statistics , two real-valued random variables ,
X
{\displaystyle X}
Y
{\displaystyle Y}
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]}
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}
Y
{\displaystyle Y}
if and only if
E
[
X
Y
]
=
0
{\displaystyle \operatorname {E} [XY]=0}
If
X
{\displaystyle X}
Y
{\displaystyle Y}
independent , with finite second moments , then they are uncorrelated. However, not all uncorrelated variables are independent.[ 1] : p. 155
Definition for two real random variables [ edit ] Two random variables
X
,
Y
{\displaystyle X,Y}
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])]}
[ 1] : p. 153 [ 2] : p. 121
X
,
Y
uncorrelated
⟺
E
[
X
Y
]
=
E
[
X
]
⋅
E
[
Y
]
{\displaystyle X,Y{\text{ uncorrelated}}\quad \iff \quad \operatorname {E} [XY]=\operatorname {E} [X]\cdot \operatorname {E} [Y]}
Definition for two complex random variables [ edit ] Two complex random variables
Z
,
W
{\displaystyle Z,W}
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])}}]}
J
Z
W
=
E
[
(
Z
−
E
[
Z
]
)
(
W
−
E
[
W
]
)
]
{\displaystyle \operatorname {J} _{ZW}=\operatorname {E} [(Z-\operatorname {E} [Z])(W-\operatorname {E} [W])]}
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 [ edit ] A set of two or more random variables
X
1
,
…
,
X
n
{\displaystyle X_{1},\ldots ,X_{n}}
autocovariance matrix
K
X
X
{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
random vector
X
=
[
X
1
…
X
n
]
T
{\displaystyle \mathbf {X} =[X_{1}\ldots X_{n}]^{\mathrm {T} }}
K
X
X
=
cov
[
X
,
X
]
=
E
[
(
X
−
E
[
X
]
)
(
X
−
E
[
X
]
)
)
T
]
=
E
[
X
X
T
]
−
E
[
X
]
E
[
X
]
T
{\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}}
Examples of dependence without correlation [ edit ] Let
X
{\displaystyle X}
Let
Y
{\displaystyle Y}
independent of
X
{\displaystyle X}
Let
U
{\displaystyle U}
U
=
X
Y
{\displaystyle U=XY}
The claim is that
U
{\displaystyle U}
X
{\displaystyle X}
Proof:
Taking into account that
E
[
U
]
=
E
[
X
Y
]
=
E
[
X
]
E
[
Y
]
=
E
[
X
]
⋅
0
=
0
,
{\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
X
{\displaystyle X}
Y
{\displaystyle Y}
cov
[
U
,
X
]
=
E
[
(
U
−
E
[
U
]
)
(
X
−
E
[
X
]
)
]
=
E
[
U
(
X
−
1
2
)
]
=
E
[
X
2
Y
−
1
2
X
Y
]
=
E
[
(
X
2
−
1
2
X
)
Y
]
=
E
[
(
X
2
−
1
2
X
)
]
E
[
Y
]
=
0
{\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,
U
{\displaystyle U}
X
{\displaystyle X}
Independence of
U
{\displaystyle U}
X
{\displaystyle X}
a
{\displaystyle a}
b
{\displaystyle b}
Pr
(
U
=
a
∣
X
=
b
)
=
Pr
(
U
=
a
)
{\displaystyle \Pr(U=a\mid X=b)=\Pr(U=a)}
a
=
1
{\displaystyle a=1}
b
=
0
{\displaystyle b=0}
Pr
(
U
=
1
∣
X
=
0
)
=
Pr
(
X
Y
=
1
∣
X
=
0
)
=
0
{\displaystyle \Pr(U=1\mid X=0)=\Pr(XY=1\mid X=0)=0}
Pr
(
U
=
1
)
=
Pr
(
X
Y
=
1
)
=
1
/
4
{\displaystyle \Pr(U=1)=\Pr(XY=1)=1/4}
Thus
Pr
(
U
=
1
∣
X
=
0
)
≠
Pr
(
U
=
1
)
{\displaystyle \Pr(U=1\mid X=0)\neq \Pr(U=1)}
U
{\displaystyle U}
X
{\displaystyle X}
Q.E.D.
If
X
{\displaystyle X}
uniformly distributed on
[
−
1
,
1
]
{\displaystyle [-1,1]}
Y
=
X
2
{\displaystyle Y=X^{2}}
X
{\displaystyle X}
Y
{\displaystyle Y}
X
{\displaystyle X}
Y
{\displaystyle Y}
Y
{\displaystyle Y}
X
{\displaystyle X}
f
X
(
t
)
=
1
2
I
[
−
1
,
1
]
;
f
Y
(
t
)
=
1
2
t
I
]
0
,
1
]
{\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,
f
X
,
Y
{\displaystyle f_{X,Y}}
0
<
X
<
Y
<
1
{\displaystyle 0<X<Y<1}
f
X
×
f
Y
{\displaystyle f_{X}\times f_{Y}}
f
X
,
Y
(
X
,
Y
)
≠
f
X
(
X
)
×
f
Y
(
Y
)
{\displaystyle f_{X,Y}(X,Y)\neq f_{X}(X)\times f_{Y}(Y)}
E
[
X
]
=
1
−
1
4
=
0
;
E
[
Y
]
=
1
3
−
(
−
1
)
3
3
×
2
=
1
3
{\displaystyle E[X]={{1-1} \over 4}=0;E[Y]={{1^{3}-(-1)^{3}} \over {3\times 2}}={1 \over 3}}
C
o
v
[
X
,
Y
]
=
E
[
(
X
−
E
[
X
]
)
(
Y
−
E
[
Y
]
)
]
=
E
[
X
3
−
X
3
]
=
1
4
−
(
−
1
)
4
4
×
2
=
0
{\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.
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 ).[ 3] jointly normally distributed random variables are independent if they are uncorrelated,[ 4] joint distribution is not joint normal (see Normally distributed and uncorrelated does not imply independent ).
Two random vectors
X
=
(
X
1
,
…
,
X
m
)
T
{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{T}}
Y
=
(
Y
1
,
…
,
Y
n
)
T
{\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{T}}
E
[
X
Y
T
]
=
E
[
X
]
E
[
Y
]
T
{\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
K
X
Y
{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}
[ 5] : p.337
Two complex random vectors
Z
{\displaystyle \mathbf {Z} }
W
{\displaystyle \mathbf {W} }
uncorrelated if their cross-covariance matrix and their pseudo-cross-covariance matrix is zero, i.e. if
K
Z
W
=
J
Z
W
=
0
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {J} _{\mathbf {Z} \mathbf {W} }=0}
where
K
Z
W
=
E
[
(
Z
−
E
[
Z
]
)
(
W
−
E
[
W
]
)
H
]
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{\mathrm {H} }]}
and
J
Z
W
=
E
[
(
Z
−
E
[
Z
]
)
(
W
−
E
[
W
]
)
T
]
{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {W} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{\mathrm {T} }]}
Two stochastic processes
{
X
t
}
{\displaystyle \left\{X_{t}\right\}}
{
Y
t
}
{\displaystyle \left\{Y_{t}\right\}}
uncorrelated if their cross-covariance
K
X
Y
(
t
1
,
t
2
)
=
E
[
(
X
(
t
1
)
−
μ
X
(
t
1
)
)
(
Y
(
t
2
)
−
μ
Y
(
t
2
)
)
]
{\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]}
[ 2] : p. 142
{
X
t
}
,
{
Y
t
}
uncorrelated
:
⟺
∀
t
1
,
t
2
:
K
X
Y
(
t
1
,
t
2
)
=
0
{\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}
^ a b Papoulis, Athanasios (1991). Probability, Random Variables and Stochastic Processes . MCGraw Hill. ISBN 0-07-048477-5 ^ a b Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3
^ Virtual Laboratories in Probability and Statistics: Covariance and Correlation , item 17.^ Bain, Lee; Engelhardt, Max (1992). "Chapter 5.5 Conditional Expectation". Introduction to Probability and Mathematical Statistics (2nd ed.). pp. 185– 186. ISBN 0534929303 ^ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers . Cambridge University Press. ISBN 978-0-521-86470-1
![{\displaystyle \operatorname {cov} [X,Y]=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ded56b10a565c8b572e646b2814a031b94770228)
![{\displaystyle \operatorname {E} [XY]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c753ab2a1aa3909b11570f26c655322dde887e)
![{\displaystyle \operatorname {Cov} [X,Y]=\operatorname {E} [(X-\operatorname {E} [X])(Y-\operatorname {E} [Y])]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94189419f6e1e2bc5a360a2246c74e939fc77b78)
![{\displaystyle X,Y{\text{ uncorrelated}}\quad \iff \quad \operatorname {E} [XY]=\operatorname {E} [X]\cdot \operatorname {E} [Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/562b87f28ef70f83e052ebec97459a5044a1125d)
![{\displaystyle \operatorname {K} _{ZW}=\operatorname {E} [(Z-\operatorname {E} [Z]){\overline {(W-\operatorname {E} [W])}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cda416fcd9c1e6ec5a2a6c995f9ab39847f7fb8)
![{\displaystyle \operatorname {J} _{ZW}=\operatorname {E} [(Z-\operatorname {E} [Z])(W-\operatorname {E} [W])]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6391f10a0cf1eb09f5da09cc83d442306f0105e3)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb7843711417e69eebdf5ce57687de2005c0abc9)
![{\displaystyle \mathbf {X} =[X_{1}\ldots X_{n}]^{\mathrm {T} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad0d84e3ae0c602a3d820b1a756134d1d1202b5)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fcbaa6c2613279e7018b348dac19099d0ba5d94)
![{\displaystyle \operatorname {E} [U]=\operatorname {E} [XY]=\operatorname {E} [X]\operatorname {E} [Y]=\operatorname {E} [X]\cdot 0=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4da767a2ab5ae8e62c4e76439b4a30a4ba624a)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d8e5b797ac2f8316c71571d441696d0bfbe8f8)
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
![{\displaystyle f_{X}(t)={1 \over 2}I_{[-1,1]};f_{Y}(t)={1 \over {2{\sqrt {t}}}}I_{]0,1]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e77c0f6e695cb006ffbc6056f809b5bd13045399)
![{\displaystyle E[X]={{1-1} \over 4}=0;E[Y]={{1^{3}-(-1)^{3}} \over {3\times 2}}={1 \over 3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da9300036cffa01c1fb604ce320a1ef8d71a0611)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e174cb7a257d296619dd6aaf727484520c2f5eb6)
![{\displaystyle \operatorname {E} [\mathbf {X} \mathbf {Y} ^{T}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a42871987cbf77a6e3ecf8ecbb23e4f33912ebcc)
![{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{\mathrm {H} }]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/686b575db50d74be88fa21e2778665659fcbc712)
![{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {W} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{\mathrm {T} }]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/068d7436f0c3f72da7d43d1459c435f2a448ac71)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf123985d84cdc25587546891caea6d23dc4f6d)
