Markov’s Inequality

18 Dec, 2012 at 20:02 | Posted in Statistics & Econometrics | 2 Comments

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 a 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 suprising today as thirty 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, s 129]


  1. That only helps in situations where the expectation value is known within some bounds;

    Since no real-life distributed function comes with accurate error propagation, the primary lesson is that no “organized” system can scale up without adequately scaling distributed feedback, parsing and testing.

    That, in turn, only means that auto-catalysis tracks discovery of new methods. Duh!

    It’s the return-on-[total]coordination, silly … but we always shy away from that simple fact. Minus total-system feedback, there’s always the temptation to think that the probability of some local projection is different this time.

    • True, but in its simplicity still beautiful and powerful when applicable 🙂

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