Editing Chebyshev's inequality (section)

==Proof==

[[Markov's inequality]] states that for any non-negative real-valued random variable ''Y'' and any positive number ''a'', we have <math>\Pr(|Y| \geq a) \leq \mathbb{E}[|Y|]/a</math>. One way to prove Chebyshev's inequality is to apply Markov's inequality to the random variable <math>Y = (X - \mu)^2</math> with <math>a = (k \sigma)^2</math>:
:<math> \Pr(|X - \mu| \geq k\sigma) = \Pr((X - \mu)^2 \geq k^2\sigma^2) \leq \frac{\mathbb{E}[(X - \mu)^2]}{k^2\sigma^2} = \frac{\sigma^2}{k^2\sigma^2} = \frac{1}{k^2}. </math>

It can also be proved directly using [[conditional expectation]]:
:<math>\begin{align}
\sigma^2&=\mathbb{E}[(X-\mu)^2]\\[5pt]
&=\mathbb{E}[(X-\mu)^2\mid k\sigma\leq |X-\mu|]\Pr[k\sigma\leq|X-\mu|]+\mathbb{E}[(X-\mu)^2\mid k\sigma>|X-\mu|]\Pr[k\sigma>|X-\mu|] \\[5pt]
&\geq(k\sigma)^2\Pr[k\sigma\leq|X-\mu|]+0\cdot\Pr[k\sigma>|X-\mu|] \\[5pt]
&=k^2\sigma^2\Pr[k\sigma\leq|X-\mu|]
\end{align}</math>
Chebyshev's inequality then follows by dividing by ''k''<sup>2</sup>''&sigma;''<sup>2</sup>.
This proof also shows why the bounds are quite loose in typical cases: the conditional expectation on the event where |''X''&nbsp;−&nbsp;''&mu;''|&nbsp;<&nbsp;''k&sigma;'' is thrown away, and the lower bound of ''k''<sup>2</sup>''&sigma;''<sup>2</sup> on the event |''X''&nbsp;−&nbsp;''&mu;''|&nbsp;&ge;&nbsp;''k&sigma;'' can be quite poor.

Chebyshev's inequality can also be obtained directly from a simple comparison of areas, starting from the representation of an expected value as the difference of two improper Riemann integrals ([[Expected value#EX as difference of integrals|last formula]] in the [[Expected value#Arbitrary real-valued random variables|definition of expected value for arbitrary real-valued random variables]]).<ref>{{cite book |last1=Uhl |first1=Roland |title=Charakterisierung des Erwartungswertes am Graphen der Verteilungsfunktion |trans-title=Characterization of the expected value on the graph of the cumulative distribution function |date=2023 |publisher=Technische Hochschule Brandenburg |doi=10.25933/opus4-2986 |doi-access=free |url=https://opus4.kobv.de/opus4-fhbrb/files/2986/Uhl2023.pdf}} p.&nbsp;5.</ref>