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Chebyshev’s Inequality

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International Encyclopedia of Statistical Science

Chebyshev’s inequality is one of the most common inequalities used in probability theory to bound the tail probabilities of a random variable X having finite variance σ2 = { Var}X. It states that

$$\mathbb{P}(\vert X - \mu \vert \geq t)\ \leq \ \frac{{\sigma }^{2}} {{t}^{2}} \quad \text{ for all }t> 0,$$
(1)

where μ = eX denotes the mean of X. Of course, the given bound is of use only if t is bigger than the standard deviation σ. Instead of proving (1) we will give a proof of the more general Markov’s inequality which states that for any nondecreasing function g : [0, ∞) → [0, ∞) and any nonnegative random variable Y

$$\mathbb{P}(Y \geq t)\ \leq \ \frac{\mathrm{e}g(Y )} {g(t)} \quad \text{ for all }t> 0.$$
(2)

Indeed, choosing Y = | X − μ | and g(x) = x 2 gives (1). The proof of Markov’s inequality is very easy: For any t > 0,

$$\begin{array}{rlrlrl} \mathbb{P}(Y \geq t) & = \int \nolimits \nolimits {1}_{\{Y \geq t\}}d\mathbb{P} \leq {\int \nolimits \nolimits }_{\{Y \geq t\}}\frac{g(Y...

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References and Further Reading

  • Chebyshev P (1867) Des valeurs moyennes. Liouville’s J Math Pure Appl 12:177–184

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  • Feller W (1971) An introduction to probability theory and its applications, vol II, 2nd edn. Wiley, New York

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  • Monhor D (2007) A Chebyshev inequality for multivariate normal distribution. Prob Eng Inform Sci 21(2):289–300

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  • Olkin I, Pratt JW (1958) A multivariate Tchebycheff inequality. Ann Math Stat 29:226–234

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  • Sellke TM, Sellke SH (1997) Chebyshev inequalities for unimodal distributions. Am Stat 51(1):34–40

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  • Vysočanskiĭ DF, Petunĭn JĪ (1979) Proof of the 3σ rule for unimodal distributions. Teor Veroyatnost i Mat Stat 21:23–35

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Author information

Authors and Affiliations

  1. Institut für Mathematische Statistik, Einsteinstraße 62, D-48149, Münster, Germany

    Gerold Alsmeyer

Authors
  1. Gerold Alsmeyer
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Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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© 2011 Springer-Verlag Berlin Heidelberg

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Alsmeyer, G. (2011). Chebyshev’s Inequality. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_167

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