What is the variance of a constant?

Question

Given a constant $c$, I know that $\text{E}(c)=c$, but what about the variance of $c$?

Answer 1

You have observed that, for any constant $\alpha$, $E[\alpha]=\alpha$, then we have $$ \operatorname{Var}(c)=\operatorname{E}[c^2]-(\operatorname{E}[c])^2=c^2-c^2=0. $$ One may also just go back to the definition of the variance, $$ \operatorname{Var}(X) = \operatorname{E}\left[(X - \operatorname{E}\left[X \right])^2 \right] $$ giving

$$ \operatorname{Var}(c)= \operatorname{E}\left[(c - c)^2 \right]=\operatorname{E}\left[0 \right]=0. $$