A very simple proof of the multivariate Chebyshev's inequality

Jorge Navarro

Communications in Statistics - Theory and Methods, 2016, vol. 45, issue 12, 3458-3463

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.

Date: 2016
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Citations: View citations in EconPapers (4)

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DOI: 10.1080/03610926.2013.873135

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