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A very simple proof of the multivariate Chebyshev's inequality

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  • Jorge Navarro

Abstract

In this short note, a very simple proof of the Chebyshev's inequality for random vectors is given. This inequality provides a lower bound for the percentage of the population of an arbitrary random vector X with finite mean μ = E(X) and a positive definite covariance matrix V = Cov(X) whose Mahalanobis distance with respect to V to the mean μ is less than a fixed value. The main advantage of the proof is that it is a simple exercise for a first year probability course. An alternative proof based on principal components is also provided. This proof can be used to study the case of a singular covariance matrix V.

Suggested Citation

  • Jorge Navarro, 2016. "A very simple proof of the multivariate Chebyshev's inequality," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(12), pages 3458-3463, June.
    • Handle: RePEc:taf:lstaxx:v:45:y:2016:i:12:p:3458-3463
    DOI: 10.1080/03610926.2013.873135
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    Cited by:

    1. Budny Katarzyna, 2019. "Power Generalization Of Chebyshev’S Inequality – Multivariate Case," Statistics in Transition New Series, Polish Statistical Association, vol. 20(3), pages 155-170, September.
    2. Budny, Katarzyna, 2022. "Improved probability inequalities for Mardia’s coefficient of kurtosis," Statistics & Probability Letters, Elsevier, vol. 191(C).
    3. Navarro Jorge, 2020. "Bivariate box plots based on quantile regression curves," Dependence Modeling, De Gruyter, vol. 8(1), pages 132-156, January.

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