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(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) $Cov(X,Y)=Cov(Y,X)$ (b) $Cov(X,X)=Var(X)$ (c) For any constant $a,Cov(aX,Y)=aCov(X,Y)$ (d) For any constant $a,Cov(a+X,Y)=Cov(X,Y)$ (e) If $X$ and $Y$ are independent, $Cov(X,Y)=0$ (f) $Cov(X,Y)=0$ does not imply that $X$ and $Y$ are independent (g) $Cov(i=1?n?a_{i}X_{i},j=1?m?b_{j}Y_{j})=i=1?n?j=1?m?a_{i}b_{j}Cov(X_{i},Y_{j})$, where $a_{i},b_{j}$ are constants

Answer :As per the data :Part A)Cov(X,Y) = Cov(Y,X)Now, Cov(X,Y) = E(XY) - E(X) E(Y)= E(YX) - E(Y) E(X)Cov(X,Y) = Cov(Y,X)