(1)

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

(2)

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

(3)

\begin{align*} V\left(\sum_{i=1}^{n}a_{i}X_{i}\right) & =E\left(\left(\sum_{i=1}^{n}a_{i}X_{i}\right)^{2}\right)-E^{2}\left(\sum_{i=1}^{n}a_{i}X_{i}\right)\\ & =E\left(\sum_{i,j}a_{i}a_{j}X_{i}X_{j}\right)-\left(\sum_{i=1}^{n}a_{i}E\left(X_{i}\right)\right)^{2}\\ & =\sum_{i,j}a_{i}a_{j}E\left(X_{i}X_{j}\right)-\sum_{i,j}a_{i}a_{j}E\left(X_{i}\right)E\left(X_{j}\right)\\ & =\sum_{i,j}a_{i}a_{j}\left(E\left(X_{i}X_{j}\right)-E\left(X_{i}\right)E\left(X_{j}\right)\right)\\ & =\sum_{i,j}a_{i}a_{j}Cov\left(X_{i},X_{j}\right) \end{align*}

(4)

(3)より、
\begin{align*} V(a_{1}X_{1}+a_{2}X_{2}) & =a_{1}a_{1}Cov\left(X_{1},X_{1}\right)+a_{2}a_{2}Cov\left(X_{2},X_{2}\right)+2a_{1}a_{2}Cov\left(X_{1},X_{2}\right)\\ & =a_{1}{}^{2}V(X_{1})+a_{2}{}^{2}V(X_{2})+2a_{1}a_{2}Cov(X_{1},X_{2}) \end{align*}

(5)

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