Proof : Let X be a random variable and a be any constant.

\begin{eqnarray*}V[aX]&=&E[\left(aX-E[aX]\right)^{2}]\\&=&E[\left(aX-aE[X]\right)^{2}]\ \ \ (\text{∵}\ \ E[aX]=aE[X]\ \ )\\&=&E[\{a\left(X-E[X]\right)\}^{2}]\\&=&E[a^{2}\left(X-E[X]\right)^{2}]\\&=&a^{2}E[\left(X-E[X]\right)^{2}]\ \ \ (\text{∵}\ \ E[a]=a\ \ )\\&=&a^{2}V[X]\end{eqnarray*}