CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The second-moment lower bound for positive probability

Statement

Let $X$ be a nonnegative square-integrable real random variable.

If $E [X^{2}] > 0$ , then $P (X > 0) \geq \frac{E [X]^{2}}{E [X^{2}]} .$
If $E [X^{2}] = 0$ , then $X = 0$ almost surely and $P (X > 0) = 0$ .

Facts & Assumptions

Given: A nonnegative square-integrable real random variable $X$ .

[L1]

The expectation of an indicator is the probability of its event (The expectation of an indicator is the probability of the event).

[L2]

Cauchy-Schwarz holds for square-integrable random variables (Cauchy-Schwarz for random variables).

[L3]

Square-integrability means that the second moment $E [X^{2}]$ is finite (Moments, variance, and covariance on a probability space).

Proof

technique · direct

1.1

The identity $X = X 1_{{X > 0}}$ holds pointwise because $X \geq 0$ . Applying [L2] to $U = X 1_{{X > 0}}$ and $V = 1_{{X > 0}}$ gives $E [X]^{2} \leq E [X^{2}] E [1_{{X > 0}}] .$ By [L1], this is $E [X]^{2} \leq E [X^{2}] P (X > 0) .$

L1L2L3

2.1

If $E [X^{2}] > 0$ , divide the inequality in step 1.1 by that positive number. If $E [X^{2}] = 0$ , then step 1.1 forces $E [X] = 0$ as well, and since $X^{2} \geq 0$ , zero second moment means $X^{2} = 0$ almost surely, hence $X = 0$ almost surely and $P (X > 0) = 0$ .

step 1.1L3algebra

3.1

Step 2.1 proves both the positive-second-moment case and the zero boundary case.

step 2.1∎

Depends on

Used by

The general inequalities compare cleanly with the published finite ones Remark

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. R. Norris, Probability and Measure, Section 4.2 (standard reference, not scraped)