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Monotonicity of Expectation

Theorem

Let XX and YY be random variables with finite expectations. If X≤YX\le Y almost surely (that is, P(X≤Y)=1\mathbb{P}(X\le Y)=1), then monotonicity of expectation states that

E[X]≤E[Y].\mathbb{E}[X]\le \mathbb{E}[Y].

Interpretation

Monotonicity says that if one random variable is never larger than another, its average is not larger either: taking expectations preserves the inequality X≤YX\le Y.

Write D=Y−XD=Y-X. The hypothesis says that D≥0D\ge0 with probability one, and an average of nonnegative values cannot be negative; by linearity of expectation, E[Y]−E[X]=E[D]≥0\mathbb{E}[Y]-\mathbb{E}[X]=\mathbb{E}[D]\ge0. An event of probability zero cannot affect any average, which is why the comparison need only hold with probability one. The converse fails: for a fair die roll XX and Y=7−XY=7-X the means are equal, yet X>YX>Y whenever X≥4X\ge4.

The theorem is mostly used to bound an expectation by a simpler one. If X≤cX\le c for a constant cc, then E[X]≤c\mathbb{E}[X]\le c. For X≥0X\ge0 and c>0c>0, the inequality c 𝟙1{X≥c}≤Xc\,\indicator_{\{X\ge c\}}\le X together with the fundamental bridge gives c P(X≥c)≤E[X]c\,\mathbb{P}(X\ge c)\le\mathbb{E}[X]: a nonnegative variable with a small mean rarely takes large values.

Key Results

  • If X≤YX\le Y almost surely, then E[X]≤E[Y]\mathbb{E}[X]\le \mathbb{E}[Y] when expectations exist.
  • Strict inequality is not guaranteed unless the ordering is strict with positive probability.
  • Monotonicity is often used to bound expectations.

Examples

  • If a premium service always costs at least as much as a basic service, its expected cost is at least as large.
  • If one policy always pays no more than another, its expected payout is no larger.
  • Truncating losses from above produces a random variable with no larger expectation.

Prerequisites

  • Linearity of Expectation: if integrable variables satisfy X≤YX\le Y almost surely, then E[Y]−E[X]=E[Y−X]≥0\mathbb{E}[Y]-\mathbb{E}[X]=\mathbb{E}[Y-X]\ge0.
  • LOTUS: computes the expectations being compared from their distributions.
  • Almost Surely

Sources

Aliases: expectation monotonicity Keywords: monotone expectation

Last updated: 20 October 2026 03:02