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I was looking at the following definition of Chebyshev's inequality
$$P(|X - E(X)| \geq r) \leq \frac{Var(X)}{r^2}$$
which includes the expected value and variance of $X$, and then I discovered there's another equivalent Chebyshev's inequality, which involves the standard deviation $\sigma$
$$...
Is there a lower bound inequality for $P(X<\mu-\sigma)$, where $P$, $\mu$ and $\sigma$ might be arbitrary but finite. For example by the Cantelli's inequality we have an upper bound $P(X<\mu-\sigma)\le0.5$. Is there also a nontrivial lower bound inequality?
