Kovarianca (oznaka C o v ( X , Y ) {\displaystyle \operatorname {Cov(X,Y)} \,} σ X Y {\displaystyle \sigma _{XY}\,} varianca , ki pa opisuje spreminjanje dveh spremenljivk, ki sta identični.
Definicija
uredi
Kovarianca med dvema realnima slučajnima spremenljivkama X {\displaystyle X\,} Y {\displaystyle Y\,} drugim momentom , je določena kot:
Cov ( X , Y ) = E [ ( X − E [ X ] ) ( Y − E [ Y ] ) ] , {\displaystyle \operatorname {Cov} (X,Y)=\operatorname {E} {\big [}(X-\operatorname {E} [X])(Y-\operatorname {E} [Y]){\big ]}\!\,,} kjer je:
E ( X ) {\displaystyle E(X)\,} pričakovana vrednost spremenljivke X {\displaystyle X\,} X {\displaystyle X\,} m × 1 {\displaystyle m\times 1\,} Y {\displaystyle Y\,} n × 1 {\displaystyle n\times 1\,} Zgornji obrazec lahko zapišemo tudi kot:
Cov ( X , Y ) = E [ X Y ] − E [ X ] ⋅ E [ Y ] . {\displaystyle \operatorname {Cov} (X,Y)=\operatorname {E} {\big [}XY{\big ]}-\operatorname {E} [X]\cdot \operatorname {E} [Y]\!\,.} Slučajne spremenljivke, ki imajo kovarianco enako 0, so nekorelirane .
Merska enota za kovarianco je enaka zmnožku merske enote za prvo slučajno spremenljivko in merske enote za drugo slučajno spremenljivko. Korelacija pa je brezrazsežna .
Značilnosti
uredi
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)\,} kjer so (velja za vse značilnosti):
X , Y , W , V {\displaystyle X,Y,W,V\,} a , b , c , d {\displaystyle a,b,c,d\,} konstante (niso slučajne spremenljivke)V a r ( X ) {\displaystyle \operatorname {Var(X)} \,} varianca Za zaporedje X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}\,} Y 1 , … , Y n {\displaystyle Y_{1},\dots ,Y_{n}\,}
Cov ( ∑ i = 1 n X i , ∑ j = 1 m Y j ) = ∑ i = 1 n ∑ j = 1 m Cov ( X i , Y j ) . {\displaystyle \operatorname {Cov} \left(\sum _{i=1}^{n}{X_{i}},\sum _{j=1}^{m}{Y_{j}}\right)=\sum _{i=1}^{n}{\sum _{j=1}^{m}{\operatorname {Cov} \left(X_{i},Y_{j}\right)}}\!\,.} Velja pa tudi:
Var ( ∑ i = 1 n a i X i ) = ∑ i = 1 n a i 2 Var ( X i ) + 2 ∑ i , j : i < j a i a j Cov ( X i , X j ) . {\displaystyle \operatorname {Var} \left(\sum _{i=1}^{n}a_{i}X_{i}\right)=\sum _{i=1}^{n}a_{i}^{2}\operatorname {Var} (X_{i})+2\sum _{i,j\,:\,i<j}a_{i}a_{j}\operatorname {Cov} (X_{i},X_{j})\!\,.} kjer so:
a 1 , … a n {\displaystyle a_{1},\dots a_{n}\,} konstante .Kadar sta X {\displaystyle X\,} Y {\displaystyle Y\,}
E ( X ⋅ Y ) = E ( X ) ⋅ E ( Y ) . {\displaystyle \operatorname {E} (X\cdot Y)=E(X)\cdot E(Y)\!\,.} Velja tudi Cauchy-Schwarzeva neenakost :
| Cov ( X , Y ) | ≤ Var ( X ) Var ( Y ) . {\displaystyle |\operatorname {Cov} (X,Y)|\leq {\sqrt {\operatorname {Var} (X)\operatorname {Var} (Y)}}\!\,.} Zunanje povezave
uredi
![{\displaystyle \operatorname {Cov} (X,Y)=\operatorname {E} {\big [}(X-\operatorname {E} [X])(Y-\operatorname {E} [Y]){\big ]}\!\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28ffaeffd311f8aea7f268dbb5e8685206db97ef)
![{\displaystyle \operatorname {Cov} (X,Y)=\operatorname {E} {\big [}XY{\big ]}-\operatorname {E} [X]\cdot \operatorname {E} [Y]\!\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3febf8b3333779a690fa7c1647a068dbd7784ed3)
