Covariance

Definition

$\mathrm{Cov}(X,Y)=E[(X-{\mu }_X)(Y-{\mu }_Y)]$

Properties

Covariance with Itself

$\mathrm{Cov}(X,X)=\mathrm{Var}(X)$

Covariance of Linear Combinations

$\mathrm{Cov}(\sum _{i=1}^n{a}_i{X}_i,\sum _{i=1}^n{b}_i{Y}_i)=\sum _{i=1}^n\sum _{j=1}^m{a}_i{b}_j\mathrm{Cov}(X_i,Y_j)$

$\mathrm{Var}(\sum _{i=1}^n{a}_i{X}_i)=\mathrm{Cov}(\sum _{i=1}^n{a}_i{X}_i,\sum _{i=1}^n{a}_i{X}_i)=\sum _{i=1}^n{a}_i^2\mathrm{Var}(X_i)+2\sum _{i,j:i<j}{a}_i{a}_j\mathrm{Cov}(X_i,X_j)$