(Solved): (Fundamentals on covariance function) This question reviews some fundamental properties of covaria ...

(Fundamentals on covariance function) This question reviews some fundamental properties of covariance function, which is essential for computing autocovariance function. Show the following properties hold. (a) $$\mathrm{Cov}(X,Y)=\mathrm{Cov}(Y,X)$$ (b) $$\mathrm{Cov}(X,X)=\mathrm{Var}(X)$$ (c) For any constant $$a,\mathrm{Cov}(aX,Y)=a\mathrm{Cov}(X,Y)$$ (d) For any constant $$a,\mathrm{Cov}(a+X,Y)=\mathrm{Cov}(X,Y)$$ (e) If $$X$$ and $$Y$$ are independent, $$\mathrm{Cov}(X,Y)=0$$ (f) $$\mathrm{Cov}(X,Y)=0$$ does not imply that $$X$$ and $$Y$$ are independent (g) $$\mathrm{Cov}\left(\underset{i=1}{\overset{n}{?}}{a}_i{X}_i,\underset{j=1}{\overset{m}{?}}{b}_j{Y}_j\right)=\underset{i=1}{\overset{n}{?}}\underset{j=1}{\overset{m}{?}}{a}_i{b}_j\mathrm{Cov}\left(X_i,Y_j\right)$$, where $$a_i,b_j$$ are constants