Mathematical Statistics Lesson of the Day – Chebyshev’s Inequality

September 4, 2014

By Eric Cai - The Chemical Statistician

(This article was originally published at The Chemical Statistician » Statistics, and syndicated at StatsBlogs.)

The variance of a random variable $X$ is just an expected value of a function of $X$ . Specifically,

$V(X) = E[(X - \mu)^2], \ \text{where} \ \mu = E(X)$ .

Let’s substitute $(X - \mu)^2$ into Markov’s inequality and see what happens. For convenience and without loss of generality, I will replace the constant $c$ with another constant, $b^2$ .

$\text{Let} \ b^2 = c, \ b > 0. \ \ \text{Then,}$

$P[(X - \mu)^2 \geq b^2] \leq E[(X - \mu)^2] \div b^2$

$P[ (X - \mu) \leq -b \ \ \text{or} \ \ (X - \mu) \geq b] \leq V(X) \div b^2$

$P[|X - \mu| \geq b] \leq V(X) \div b^2$

Now, let’s substitute $b$ with $k \sigma$ , where $\sigma$ is the standard deviation of $X$ . (I can make this substitution, because $\sigma$ is just another constant.)

$\text{Let} \ k \sigma = b. \ \ \text{Then,}$

$P[|X - \mu| \geq k \sigma] \leq V(X) \div k^2 \sigma^2$

$P[|X - \mu| \geq k \sigma] \leq 1 \div k^2$

This last inequality is known as Chebyshev’s inequality, and it is just a special version of Markov’s inequality. In a later Statistics Lesson of the Day, I will discuss the motivation and intuition behind it. (Hint: Read my earlier lesson on the motivation and intuition behind Markov’s inequality.)

Filed under: Mathematical Statistics, Mathematics, Probability, Statistics, Statistics Lesson of the Day Tagged: Chebyshev's inequality, expectation, expected value, Markov's inequality, math, mathematical statistics, mathematics, probability, random variable, statistics, variance

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Tags: Chebyshev's inequality, expectation, expected value, Markov's inequality, Math, Mathematical Statistics, mathematics, probability, Random variable, statistics, Statistics Lesson of the Day, Variance

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