উইকিপিডিয়া, মুক্ত বিশ্বকোষ থেকে
সহভেদাঙ্ক - সহভেদাঙ্ক বা কোভ্যারিয়্যেন্স হলো এমন একটি পরিমাপ যার দ্বারা দুইটি চলকের একত্র পরিবর্তন নির্ণয় করা হয়। ভেদাঙ্ক হলো সহভেদাঙ্কের একটি বিশেষ রূপ, যখন চলক দুটি একদম একই রকম।
যদি
X
{\displaystyle X}
Y
{\displaystyle Y}
Cov
(
X
,
Y
)
=
E
[
(
X
−
E
[
X
]
)
(
Y
−
E
[
Y
]
)
]
=
E
[
X
Y
]
−
E
[
X
]
⋅
E
[
Y
]
{\displaystyle \operatorname {Cov} (X,Y)=\operatorname {E} {\big [}(X-\operatorname {E} [X])(Y-\operatorname {E} [Y]){\big ]}=\operatorname {E} {\big [}XY{\big ]}-\operatorname {E} [X]\cdot \operatorname {E} [Y]}
যেখানে
E
[
X
]
{\displaystyle \operatorname {E} [X]}
X
{\displaystyle X}
E
[
Y
]
{\displaystyle \operatorname {E} [Y]}
Y
{\displaystyle Y}
সহ-ভেদাঙ্ক
Cov
(
X
,
Y
)
{\displaystyle \operatorname {Cov} (X,Y)}
X
{\displaystyle X}
Y
{\displaystyle Y}
Cor
(
X
,
Y
)
{\displaystyle \operatorname {Cor} (X,Y)}
যদি
X
{\displaystyle X}
Y
{\displaystyle Y}
W
{\displaystyle W}
V
{\displaystyle V}
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
d
{\displaystyle d}
Cov
(
X
,
a
)
=
0
{\displaystyle \operatorname {Cov} (X,a)=0\,}
Cov
(
X
,
X
)
=
Var
(
X
)
{\displaystyle \operatorname {Cov} (X,X)=\operatorname {Var} (X)\,}
Cov
(
X
,
Y
)
=
Cov
(
Y
,
X
)
{\displaystyle \operatorname {Cov} (X,Y)=\operatorname {Cov} (Y,X)\,}
Cov
(
a
X
,
b
Y
)
=
a
b
Cov
(
X
,
Y
)
{\displaystyle \operatorname {Cov} (aX,bY)=ab\,\operatorname {Cov} (X,Y)\,}
Cov
(
X
+
a
,
Y
+
b
)
=
Cov
(
X
,
Y
)
{\displaystyle \operatorname {Cov} (X+a,Y+b)=\operatorname {Cov} (X,Y)\,}
Cov
(
a
X
+
b
Y
,
c
W
+
d
V
)
=
a
c
Cov
(
X
,
W
)
+
a
d
Cov
(
X
,
V
)
+
b
c
Cov
(
Y
,
W
)
+
b
d
Cov
(
Y
,
V
)
{\displaystyle \operatorname {Cov} (aX+bY,cW+dV)=ac\,\operatorname {Cov} (X,W)+ad\,\operatorname {Cov} (X,V)+bc\,\operatorname {Cov} (Y,W)+bd\,\operatorname {Cov} (Y,V)\,}
![{\displaystyle \operatorname {Cov} (X,Y)=\operatorname {E} {\big [}(X-\operatorname {E} [X])(Y-\operatorname {E} [Y]){\big ]}=\operatorname {E} {\big [}XY{\big ]}-\operatorname {E} [X]\cdot \operatorname {E} [Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/937976fcb06431b6ffe8d4d61893c601190acbad)
![{\displaystyle \operatorname {E} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44dd294aa33c0865f58e2b1bdaf44ebe911dbf93)
![{\displaystyle \operatorname {E} [Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/639e8577c6faffc0471c7e123ead30970034e6d5)
