Value of the variance

Question

I'm trying to prove that this statement is true (maybe it's false), but I don't know how to do it:

If all the values of a statistic variable $X$ are in an interval of length 2, then the variance of $X$ cannot be grater than 4.

Answer 1

$\newcommand{var}{\operatorname{Var}}$$\var(X)$ is most generally defined by $E((X-\mu)^2)$ where $\mu$ is $X$'s expected value. If the support of $X$ is on an interval of length $2$, the mean must obviously fall within the support (property A), so any sample from $X$ can be at most $2$ away from the mean, i.e. $|X-\mu|\le2$ and thus $0\le(X-\mu)^2\le4$. Treating $(X-\mu)^2$ as a random variable, its expected value – $\var(X)$ – by property A must be in $[0,4]$, and the claim is proved.