Chebyshev's 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

\(\ds \)

\(=\)

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

$\blacksquare$

Source of Name

This entry was named for Pafnuty Lvovich Chebyshev.

Sources

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