(Probability theory:)

A very useful elementary inequality, which can be used to prove many important inequalities (e.g. Chebyshev's inequality). Markov's inequality is the trite saying that if the average height is under 2 metres, then at most 10% of the population have height above 20 metres.

**LEMMA.** Let X≥0 be a random variable for which the expectation μ=**E**X exists. Then for any t≥0,

**P**(X≥tμ) ≤ 1/t

**PROOF.** If X=0 is constant, the lemma is easily true. Otherwise... define Y=tμ if X≥tμ, Y=0 otherwise. Then X≥Y, so **E**X≥**E**Y. But we can explicitly compute **E**Y, so:

μ = **E**X ≥ **E**Y = **P**(X≥tμ)tμ

Cancelling by μ, we have the lemma.