Markov’s inequality (wonkish)

20 Dec, 2019 at 10:13 | Posted in Statistics & Econometrics | 1 Comment

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]

1 Comment

  1. I wonder if you can relate this inequality to finance:
    .
    The price of an option with a 10% higher strike price will be less than or equal to the spot price divided by 10. So you can bet on an index going up 10% for one-tenth the present index price. If today’s index price is $10, you can bet it will go to $11 for $1.
    .
    The Black-Scholes equation gets basically to the same place, I believe: you can bet on future price movements for a fraction of the payoff, should the bet come in at the money. Option returns are outsized.


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