# Bienaymé-Chebyshev Inequality

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## Theorem

Let $X$ be a random variable.

Let $\expect X = \mu$ for some $\mu \in \R$.

Let $\var X = \sigma^2$ for some $\sigma^2 \in \R_{> 0}$.

Then, for all $k > 0$:

- $\map \Pr {\size {X - \mu} \ge k \sigma} \le \dfrac 1 {k^2}$

## Proof 1

Let $f$ be the function:

- $\map f x = \begin{cases} k^2 \sigma^2 & : \size {x - \mu} \ge k \sigma \\ 0 & : \text{otherwise} \end{cases}$

By construction, we see that:

- $\map f x \le \size {x - \mu}^2 = \paren {x - \mu}^2$

for all $x$.

This means that:

- $\expect {\map f X} \le \expect {\paren {X - \mu}^2}$

By definition of variance:

- $\expect {\paren {X - \mu}^2} = \var X = \sigma^2$

By definition of expectation of discrete random variable, we can show that:

\(\ds \expect {\map f X}\) | \(=\) | \(\ds k^2 \sigma^2 \map \Pr {\size {X - \mu} \ge k \sigma} + 0 \cdot \map \Pr {\size {X - \mu} \le k \sigma}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds k^2 \sigma^2 \map \Pr {\size {X - \mu} \ge k \sigma}\) |

Putting this together, we have:

\(\ds \expect {\map f X}\) | \(\le\) | \(\ds \expect {\paren {X - \mu}^2}\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds k^2 \sigma^2 \map \Pr {\size {X - \mu} \ge k \sigma}\) | \(\le\) | \(\ds \sigma^2\) |

By dividing both sides by $k^2 \sigma^2$, we get:

- $\map \Pr {\size {X - \mu} \ge k \sigma} \le \dfrac 1 {k^2}$

$\blacksquare$

## Proof 2

Note that as $k > 0$ and $\sigma > 0$, we have $k \sigma > 0$.

We therefore have:

\(\ds \map \Pr {\size {X - \mu} \ge k \sigma}\) | \(=\) | \(\ds \map \Pr {\paren {X - \mu}^2 \ge \paren {k \sigma}^2}\) | ||||||||||||

\(\ds \) | \(\le\) | \(\ds \frac {\expect {\paren {X - \mu}^2} } {\paren {k \sigma}^2}\) | as $k \sigma > 0$, we can apply Markov's Inequality: Corollary | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\sigma^2} {k^2 \sigma^2}\) | Definition of Variance | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac 1 {k^2}\) |

$\blacksquare$

## Also known as

The **Bienaymé-Chebyshev Inequality** is also known as **Chebyshev's Inequality**.

However, some sources use this name to mean the **Chebyshev's Sum Inequality**, which is a completely different result.

Hence, to avoid confusion, the name **Chebyshev's Inequality** is not used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Source of Name

This entry was named for Pafnuty Lvovich Chebyshev and Irénée-Jules Bienaymé.

## Historical Note

The result now known as the **Bienaymé-Chebyshev Inequality** was first formulated, without proof, by Irénée-Jules Bienaymé in $1853$.

The proof was provided by his friend and colleague Pafnuty Lvovich Chebyshev in $1867$.

Chebyshev's student Andrey Andreyevich Markov provided another proof in his $1884$ Ph.D. thesis.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Chebyshev's inequalities**