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  "body": "\n## Statement\n\nLet $Z \\ge 0$ be a real [[random variable|0PNRKMRC55]] with $0 \u003c E[Z^2] \u003c \\infty$, and let $\\theta \\in [0, 1]$.\nThen\n$$P(Z \u003e \\theta \\, E[Z]) \\ge (1 - \\theta)^2 \\frac{E[Z]^2}{E[Z^2]}.$$\n\n## Remarks\n\nProof: split $E[Z] = E[Z \\mathbf{1}_{\\set{Z \\le \\theta E[Z]}}] + E[Z \\mathbf{1}_{\\set{Z \u003e \\theta E[Z]}}]$.\nThe first term is at most $\\theta E[Z]$, and by the [[Cauchy–Schwarz inequality|0RFWW56Y22]] the second is at most $\\sqrt{E[Z^2] \\, P(Z \u003e \\theta E[Z])}$.\nRearranging gives the bound.\nWith $\\theta = 0$ it becomes the second moment method: $P(Z \u003e 0) \\ge E[Z]^2 / E[Z^2]$.\nExample: if $Z$ counts the triangles in an [[Erdős–Rényi random graph|6E83YKANV3]] and $E[Z^2] / E[Z]^2 \\to 1$, then $P(Z \u003e 0) \\to 1$, so a triangle exists with high probability.\nThe bound is a lower bound on the probability of being large, while [[Markov's inequality|05PWWHYBX5]] and [[Chebyshev's inequality|02HZ1BK3MW]] give upper bounds.\nThe hypothesis $Z \\ge 0$ is needed: the constant $Z = -1$ has $E[Z]^2 / E[Z^2] = 1$, but with $\\theta = 0$ the event $\\set{Z \u003e 0}$ is empty.\n\n## Explanations\n\n### L1 human\n\nTake a nonnegative quantity whose second moment is not much larger than the square of its mean.\nThen the quantity cannot be small most of the time.\nIt must exceed a fixed fraction of its mean with a probability bounded away from $0$.\n\n### L2 human\n\nThe inequality is the basis of the second moment method in combinatorics and random graphs, where one shows that a count $Z$ is positive with high probability.\nIn analysis, Paley and Zygmund used it for random series, to show that a random trigonometric series is large with positive probability.\nIt complements the first moment method, which uses $P(Z \u003e 0) \\le E[Z]$ for integer-valued $Z \\ge 0$.\n",
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