(1)

\begin{align*} Cov(X,Y) & =E\left(\left(X-E(X)\right)\left(Y-E(Y)\right)\right)\\ & =E(XY)-E\left(XE(Y)\right)-E\left(YE(X)\right)+E\left(E(X)E(Y)\right)\\ & =E(XY)-E(X)E(Y) \end{align*}

(2)

\begin{align*} V(X) & =E\left(\left(X-E(X)\right)^{2}\right)\\ & =E\left(X^{2}-2XE(X)+E^{2}(X)\right)\\ & =E\left(X^{2}\right)-2E^{2}(X)+E^{2}(X)\\ & =E\left(X^{2}\right)-E^{2}(X) \end{align*}

(2)-2

(1)で\(X=Y\)とすると、

\[ Cov(X,X)=E\left(X^{2}\right)-E^{2}(X) \]

となり、

\begin{align*} Cov(X,X) & =E\left(\left(X-E(X)\right)^{2}\right)\\ & =V(X) \end{align*}

であるので、
\[ V(X)=E\left(X^{2}\right)-E^{2}(X) \]