(Solved):   Hello. Can you please find the proof for these properties Properties of covariance and co ...

 

Hello. Can you please find the proof for these properties

Properties of covariance and correlation \[ \begin{array}{l} \operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)+2 \operatorname{Cov}(X, Y) \\ \operatorname{Cov}(a X, b Y)=a b \operatorname{Cov}(X, Y) \\ \operatorname{Var}(a X+b Y+c)=a^{2} \operatorname{Var}(X)+b^{2} \operatorname{Var}(Y)+2 a b \operatorname{Cov}(X, Y) \\ \operatorname{Cov}(X, Y)=\operatorname{Cov}(Y, X) \\ \operatorname{Cov}(X, X)=\operatorname{Var}(X) \end{array} \]