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# Chebyshev’s and Markov’s Inequality Theorems by
Lars Pålsson Syll
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Summary:
Chebyshev’s and Markov’s Inequality Theorems Chebyshev’s Inequality Theorem — named after Russian mathematician Pafnuty Chebyshev (1821-1894) — states that for a population (or sample) at most 1/k2 of the distribution’s values can be more than k standard deviations away from the mean. The beauty of the theorem is that although we may not know the exact distribution of the data — e.g. if it’s normally distributed  — we may still say with certitude (since the theorem holds universally)  that there are bounds on probabilities! Another beautiful result of probability theory is Markov’s inequality (after the Russian mathematician Andrei Markov (1856-1922)): If X is a non-negative stochastic variable (X ≥ 0) with a finite expectation value E(X), then for every

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## Chebyshev’s and Markov’s Inequality Theorems

Chebyshev’s Inequality Theorem — named after Russian mathematician Pafnuty Chebyshev (1821-1894) — states that for a population (or sample) at most 1/kof the distribution’s values can be more than k standard deviations away from the mean. The beauty of the theorem is that although we may not know the exact distribution of the data — e.g. if it’s normally distributed  — we may still say with certitude (since the theorem holds universally)  that there are bounds on probabilities!

Another beautiful result of probability theory is Markov’s inequality (after the Russian mathematician Andrei Markov (1856-1922)):

If X is a non-negative stochastic variable (X ≥ 0) with a finite expectation value E(X), then for every a > 0

P{X ≥ a} ≤ E(X)/a

If the production of cars in a factory during a week is assumed to be a stochastic variable with an expectation value (mean) of 50 units, we can — based on nothing else but the inequality — conclude that the probability that the production for a week would be greater than 100 units can not exceed 50% [P(X≥100)≤(50/100)=0.5=50%]

I still feel humble awe at this immensely powerful result. Without knowing anything else but an expected value (mean) of a probability distribution we can deduce upper limits for probabilities. The result hits me as equally surprising today as forty-five years ago when I first run into it as a student of mathematical statistics. Professor at Malmö University. Primary research interest - the philosophy, history and methodology of economics.