Suppose X is a non-negative random variable and E(X) = m. Then how to prove the probability P[X ≥ 2m] ≤ 1/2, more generally P[X ≥ cm] ≤ 1/c, where c > 1?
2026-03-25 20:36:41.1774471001
How to prove the probability of 2 times expectation is less than 1/2?
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Assuming that $P[X \geq cm] > 1/c \implies E(X) \geq cm*P[X \geq cm] > m$, since X is non-negative. This contradicts the assumption that $E(X) = m$, so it must be that $P[X \geq cm] \leq 1/c$.