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Key Inequalities in Probability

Often you do not know a distribution, only a summary of it: a mean, perhaps a variance. Inequalities let you make rigorous statements anyway. They trade precision for generality, giving a bound that holds for every distribution matching what you know.

That generality is the point. A bound that assumes normality tells you nothing when returns are fat-tailed; these hold regardless.

Markov's inequality

For a non-negative random variable XX and any a>0a > 0:

Markov's inequality
P(X≥a)≤E[X]aP(X \geq a) \leq \frac{E[X]}{a}

A bound from the mean alone, assuming nothing about the distribution. Weak, and it never fails you.

Only the mean is required. If daily trading volume averages 1 million shares, the probability of a day above 10 million is at most 110\frac{1}{10}. That is all you can say knowing only the average, and it is genuinely all: some distribution with that mean achieves it.

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