Markov Inequality

Explore how the Markov bound P(X ≥ a) ≤ 𝔼[X] / a compares to the real tail probability across different distributions.

Distribution

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Threshold a = 120 min

65100150200220

P(X ≥ a)

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Actual probability

𝔼[X] / a

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Markov upper bound

Bound / Actual

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How loose is the bound?

The Cauchy distribution has no finite mean — Markov's inequality does not apply.