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Markov's Inequality and Chebyshev's Inequality for Tail Probabilities: A Sharper Image

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  • Joel E. Cohen

Abstract

Markov's inequality gives an upper bound on the probability that a nonnegative random variable takes large values. For example, if the random variable is the lifetime of a person or a machine, Markov's inequality says that the probability that an individual survives more than three times the average lifetime in the population of such individuals cannot exceed one-third. Here we give a simple, intuitive geometric interpretation and derivation of Markov's inequality. These results lead to inequalities sharper than Markov's when information about conditional expectations is available, as in reliability theory, demography, and actuarial mathematics. We use these results to sharpen Chebyshev's tail inequality also.

Suggested Citation

  • Joel E. Cohen, 2015. "Markov's Inequality and Chebyshev's Inequality for Tail Probabilities: A Sharper Image," The American Statistician, Taylor & Francis Journals, vol. 69(1), pages 5-7, February.
    • Handle: RePEc:taf:amstat:v:69:y:2015:i:1:p:5-7
    DOI: 10.1080/00031305.2014.975842
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    References listed on IDEAS

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    1. Liang Hong, 2012. "A Remark on the Alternative Expectation Formula," The American Statistician, Taylor & Francis Journals, vol. 66(4), pages 232-233, November.
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