کوواریانس

${\displaystyle \operatorname {Cov} (X,a)=0\,}$

${\displaystyle \operatorname {Cov} (X,X)=\operatorname {Var} (X)\,}$

${\displaystyle \operatorname {Cov} (X,Y)=\operatorname {Cov} (Y,X)\,}$

${\displaystyle \operatorname {Cov} (aX,bY)=ab\,\operatorname {Cov} (X,Y)\,}$

${\displaystyle \operatorname {Cov} (X+a,Y+b)=\operatorname {Cov} (X,Y)\,}$

${\displaystyle \operatorname {Cov} (aX+bY,cW+dV)=ac\,\operatorname {Cov} (X,W)+ad\,\operatorname {Cov} (X,V)+bc\,\operatorname {Cov} (Y,W)+bd\,\operatorname {Cov} (Y,V)\,}$

${\displaystyle \operatorname {Cov} (X,Y)=\operatorname {E} \left((X-\operatorname {E} (X))\cdot (Y-\operatorname {E} (Y))\right)=\operatorname {E} (X\cdot Y)-\operatorname {E} (X)\cdot \operatorname {E} (Y)}$

${\displaystyle \sigma ^{2}\left(\sum _{i=1}^{n}a_{i}X_{i}\right)=\sum _{i=1}^{n}a_{i}^{2}\sigma ^{2}(X_{i})+2\sum _{i,j\,:\,i<j}a_{i}a_{j}\operatorname {cov} (X_{i},X_{j})=\sum _{i,j}{a_{i}a_{j}\operatorname {cov} (X_{i},X_{j})}}$

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