= Truncated first moment bound
{title2=$\mathbb E[X\mathbf1_{\{X>\theta\mathbb E X\}}]\geq(1-\theta)\mathbb E X$}
For a nonnegative <random variable> with finite mean $\mu$ and $0\leq\theta\leq1$, the part of the <expectation> on $X\leq\theta\mu$ is at most $\theta\mu$. Subtracting gives $\mathbb E[X\mathbf1_{\{X>\theta\mu\}}]\geq(1-\theta)\mu$. If the <second moment> is positive and finite, applying the <Cauchy-Schwarz inequality> to $X$ and the <indicator variable> gives the <Paley-Zygmund inequality> $\mathbb P(X>\theta\mu)\geq(1-\theta)^2\mu^2/\mathbb E[X^2]$.
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