Chebychev's inequality over a discrete random variable: $<$ vs $\le$ By Chebychev's inequality it holds that
$$
\begin{split}
Pr(|X-\mu |<\epsilon )>1-{\frac {\sigma ^{2}}{\epsilon ^{2}}} \\
\end{split}
$$
For a discrete random variable $X$, does the following hold?
$$
\begin{split}
Pr(|X-\mu | \le \epsilon )>1-{\frac {\sigma ^{2}}{\epsilon ^{2}}} \\
\end{split}
$$
Best regards

Well, the Chebyshev's inequality is
$$P(|X- \mu|\geq\epsilon)\leq \frac{\sigma^2}{\epsilon^2}$$
In words, it says that if the variance
is small, then the random variable is unlikely to fall
too far off from the mean.
