I came across a claim in a paper that I think reduces to:
Given a real valued, zero mean random variable $X$ with finite second moment and a constant $t$:
$$\mathbb P ( |X| > \epsilon ) \leq \frac{1}{\epsilon^2}\mathbb E \left [ X^2 \mathbb 1 \{ X^2 > \epsilon^2\} \right ]$$
Which looks like Chebychev's inequality, except using a truncated second moment, rather than the variance. My proof:
$$\mathbb P ( X^2 > \epsilon ) = \mathbb P ( X^2 \mathbb 1 \{|X| > \epsilon\} > \epsilon ) \leq \frac{1}{\epsilon}\mathbb E\{X^2\mathbb 1 \{|X| > \epsilon\}\}$$
My question:
Is this result correct? Does it have a name?