Covariance

Description:

• To find the “correlation” between 2 variables, and Correlation is the standard value to easier understood
• If $X$ and $Y$ are independent, then their covariance, $\text{Cov}(X,Y)=0$
• But $\text{Cov}(X,Y)=0$ doesn’t mean that they are independent
• $\text{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]=E[XY]-E[X]E[Y]$
  • Expected value

Propositions:

• $\text{Cov}(X,Y)=\text{Cov}(Y,X)$
• $\text{Cov}(X,X)=Var(X)$
  • Variance
• $\text{Cov}(aX,Y)=a\text{Cov}(X,Y)$
• $\text{Cov}(-X,Y)=-\text{Cov}(X,Y)$
• $\text{Cov}(\sum _{i=1}^n{X}_i,\sum _{j=1}^m{Y}_j)=\sum _{i=1}^n\sum _{j=1}^m\text{Cov}(X_i,Y_j)$
• If $Y_j=X_j,j=1,...,n$
  • $Var\left(\sum _{i=1}^n{X}_i\right)=\sum _{i=1}^{n}Var(X_i)+\sum _{i\ne j}\text{Cov}(X_i,X_j)$ $=\sum _{i=1}^{n}Var(X_i)+2\sum _{i<j}\text{Cov}(X_i,X_j)$
  • If $X_i,...,X_n$ are pairwise independent, then $Var\left(\sum _{i=1}^n{X}_i\right)=\sum _{i=1}^{n}Var(X_i)$