## Adding Constants The [[Variance]] stays unchanged if we add a constant $b$ to the [[Random Variable]] $X$. We are only changing the level, not the dispersion of the outcomes. $ \mathrm{Var}(X+b) = \mathrm{Var}(X) $ ^7759b1 **Proof:** - Let a new r.v. $Y$ be a transformed version of $X$. We can calculate its [[Expectation]]. $ \begin{align} Y&=X+b \\ \mathbb E[Y] &= \mathbb E[X+b]\\ &= \mathbb E[X]+b \\ &= \mu +b \end{align} $ - Take the standard formula for the variance and express variables in terms $X$. $ \begin{align} \mathrm{Var}(Y) &= \mathbb E\big[(Y-\mathbb E[Y])^2 \big] \\ &=\mathbb E \big[(X+ b-(\mu+ b))^2 \big] \\ &=\mathbb E \big[(X+\mu)^2 \big]\\ &= \mathrm{Var}(X) \end{align} $ ## Scaling by Factor When scaling $X$ by a factor of $a$, the variance increases by $a^2$. Big values are affected stronger by the scaling, leading to higher dispersion. $ \mathrm{Var}(aX) = a^2*\mathrm{Var}(X) $ **Proof:** - Let a new r.v. $Y$ be a transformed version of $X$. We can calculate its expectation. $ \begin{aligned} Y&=aX \\ \mathbb E[Y] &= a\mathbb E[X] = a\mu \end{aligned} $ - Take the standard formula for the variance and express variables in terms $X$. $ \begin{align} \mathrm{Var}(Y) &= \mathbb E\big[(Y-\mathbb E[Y])^2 \big] \\[2pt] &= \mathbb E\big[(aX-a\mu)^2 \big] \\[2pt] &= \mathbb E\big[(a(X-\mu))^2 \big] \\[2pt] &= \mathbb E\big[a^2(X-\mu)^2 \big] \\[2pt] &= a^2\mathbb E\big[(X-\mu)^2 \big] \\[2pt] &= a^2 \mathrm{Var}(X) \end{align} $