Cantelli inequality
= Cantelli inequality
{c}
{title2=$P(X-\mathbb EX\geq a)\leq\sigma^2/(\sigma^2+a^2)$}
= Cantelli's inequality
{c}
{synonym}
= One-sided Chebyshev inequality
{synonym}
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.