by One of the most beautiful results 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
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One of the most beautiful results 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 years ago when I first run into it as a student of mathematical statistics.
[For a derivation of the inequality, see e.g. Sheldon Ross, Introduction to Probability and Statistics for Engineers and Scientists, Academic Press, 2009]