Nonnegative random variable ($X\ge 0\text{ almost surely}$)

A nonnegative random variable is a random variable $X$ satisfyingThus $\{X\ge 0\}$ is an almost sure event. It is stronger than having a nonnegative mean, but it can allow negative values on a zero-probability event. A nonnegative mean alone permits large negative and positive outcomes to cancel.

For an original discrete example, let $X=0$ with probability $0.9$ and $X=10$ with probability $0.1$. Then $\mathbb{E}[X]=1$, even though the variable is usually zero. The discrete calculation is the sum of value times probability; the primary text defines that finite-expectation operation in its expected-value section.^[ref]

A particularly useful consequence is Markov's inequality. For $a>0$, the pointwise bound $X\ge a{1}_{\{X\ge a\}}$ holds almost surely. Taking expected values givesIn the example, $\mathrm{Pr}(X\ge 5)=0.1\le 1/5$. The bound need not be exact; it uses only nonnegativity and the mean. With an infinite nonnegative expectation it is uninformative rather than a finite tail estimate.

The same argument proves that a nonnegative random variable with zero expectation is zero almost surely: for every positive integer $n$, $\mathrm{Pr}(X\ge 1/n)=0$, and $\{X>0\}$ is the countable union of these events. Without nonnegativity this conclusion fails; equal probabilities at $-1$ and $1$ give mean zero without a zero variable.

Nonnegativity also makes the tail integral formula for expectation natural:It follows by expressing $X$ as the integral of the indicators $1_{\{X>t\}}$ and interchanging nonnegative integrals. The resulting extended expectation can be infinite. This representation connects an average amount or waiting time with the areas of its tail probabilities, rather than requiring a probability density function.