Chebyshev's inequalities give a significant information about the bounds
of a random variable with respect to its mean. They play an important role in
the proof of the weak
law of large numbers or law of averages. They are named after the famous
Russian mathematician
Pafnuty Chebyshev (1821--1894).
div id="theorem1" class="theorem">
Theorem (Chebyshev): :
Let
X be a random variable with a finite mean μ and standard
deviation σ. Then for any positive number
a,
\[
\Pr \left[ | X- \mu | \ge a \right] \le \frac{\sigma^2}{a^2} .
\]
When
a =
kσ, the inequality can be written in the
equivalent form:
\[
\Pr \left[ | X- \mu | \ge k\sigma \right] \le \frac{1}{k^2} .
\]
▣
When we need to distinguish the above inequality and a host of othehr similar
ones, it is common to call it the
\[
\Pr \left[ X \ge \mu + \varepsilon \right] \le \frac{\sigma^2}{\sigma^2 +
\varepsilon^2} , \qquad \Pr \left[ X \le \mu - \varepsilon \right] \le \frac{\sigma^2}{\sigma^2 +
\varepsilon^2} .
\]
▣
