$\operatorname{Var}X \leq \frac{1}{4}$ for any random variable $X$ bounded in $[0,1]$?

Question

I am not sure if it is trivial or if it is even true but a paper I am reading seems to use $\operatorname{Var}X \leq \frac{1}{4}$. I am fine with the obvious bound $\operatorname{Var}X \leq 1$ but curious if it can be even tighter as in $\operatorname{Var}X \leq \frac{1}{4}$. If the $\frac{1}{4}$ bound is not true, a counterexample is appreciated.

Answer 1

It is true.

$V(X)=E(X^2)-E^2(X)$. Since $0\leq X \leq 1$ we get $X^2\leq X$ so $E(X^2)\leq E(X)$ which means $V(X)\leq E(X)-E^2(X)$. The function $f(x)=x-x^2$ has a max at $x=0.5$ so $V(X)\leq f(E(X))\leq 0.25$.