Chebyshev inequality

For a random variable $X$ of finite variance and $a>0$,

For a finite-variance random variable, shift its centred version by $b>0$ and apply Markov inequality to the square. Minimizing $({\sigma }^2+b^2)/(a+b{)}^2$ at $b={\sigma }^2/a$ proves the bound. A two-point distribution attains equality. If the variance is zero, the centred variable vanishes almost surely.