协方差

「Covariance」的各地常用譯名

公式

$\operatorname {cov} (X,Y)=\operatorname {E} ((X-\mu )(Y-\nu ))=\operatorname {E} (X\cdot Y)-\mu \nu .$

$E(X\cdot Y)=E(X)\cdot E(Y)=\mu \nu ,$

$\eta ={\dfrac {\operatorname {cov} (X,Y)}{\sqrt {\operatorname {var} (X)\cdot \operatorname {var} (Y)}}}\ ,$

属性

$\operatorname {cov} (X,X)=\operatorname {var} (X)$
$\operatorname {cov} (X,Y)=\operatorname {cov} (Y,X)$
$\operatorname {cov} (aX,bY)=ab\,\operatorname {cov} (X,Y)$

$\operatorname {cov} \left(\sum _{i=1}^{n}{X_{i}},\sum _{j=1}^{m}{Y_{j}}\right)=\sum _{i=1}^{n}{\sum _{j=1}^{m}{\operatorname {cov} \left(X_{i},Y_{j}\right)}}$

$\operatorname {var} \left(\sum _{i=1}^{n}X_{i}\right)=\sum _{i=1}^{n}\operatorname {var} (X_{i})+2\sum _{i,j\,:\,i

协方差矩阵

$\operatorname {cov} (X,Y)=\operatorname {E} ((X-\mu )(Y-\nu )^{\top }).$
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