Proof of variance under scaling

Proof

We compute directly from the definition of the variance.

1. Let $X$ be a real random variable with finite variance, and let $a \in \R$.
2. By the definition of the variance, $X$ is integrable. Let $m = E[X]$. Then $E[(X - m)^2] = \operatorname{Var}(X) < \infty$.
3. By linearity of expectation, $aX$ is integrable and $E[aX] = a m$.
4. Pointwise, $(aX - am)^2 = a^2 (X - m)^2$.
5. The variable $(X - m)^2$ is integrable by step 2. By linearity of the expected value, $E[a^2 (X - m)^2] = a^2 E[(X - m)^2]$.
6. By steps 3 to 5 and the definition of the variance, $\operatorname{Var}(aX) = E[(aX - am)^2] = a^2 \operatorname{Var}(X)$.