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=EX 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 EX≥EY. But we can explicitly compute EY, so:
μ = EX ≥ EY = P(X≥tμ)tμ
Cancelling by μ, we have the lemma.