Generated Event¶

Definition¶

Let $X\colon S\to\mathbb{R}$ be a random variable and let $B\subseteq\mathbb{R}$ be a Borel set: a set built from open intervals using complements and countable unions, such as any interval or finite set. The event generated by $X$ and $B$ is

\{\omega\in S\colon X(\omega)\in B\}.

Interpretation¶

A generated event turns a statement about the value of a random variable into an event in the original sample space: " $X$ lies in $B$ " becomes the set of outcomes $\omega$ with $X(\omega)\in B$ $B$ .

A sample space with an oval region of outcomes, arrows mapping outcomes to a number line, and an interval B; the outcomes mapped into B form the event that X lies in B.

In the figure the two outcomes inside the shaded oval are sent into the interval $B$ , and the two outcomes outside it are sent to points outside $B$ . The shaded set, labelled $X^{-1}(B)$ , is the generated event: exactly the outcomes whose arrows end in $B$ . The notation does not require $X$ to have an inverse function; it only names the set of outcomes sent into $B$ .

The usual shorthand drops the outcome: $\{X\in B\}$ $B}$ , $\{X\le x\}$ and $\{X=x\}$ are sets of outcomes, and $\mathbb{P}(X\in B)$ is the probability of such a set. Different sets $B$ can generate the same event: for a die roll $X$ , the sets $\{2,4,6\}$ and $\{2,4,6,7.5\}$ both generate "an even number is rolled", because no outcome is sent to $7.5$ . Operations on sets of values carry over to events, for example $\{X\in B\}^c=\{X\notin B\}$ and $\{X\in B\}\cap\{X\in C\}=\{X\in B\cap C\}$ $C}$ , so the rules for complements and intersections of events apply directly.

Generated events connect a random variable to its distribution, which is defined by the probabilities $\mathbb{P}(X\in B)$ $B)$ , and to its CDF, built from the events $\{X\le x\}$ $x}$ . Math III later studies the same construction for general functions under the name preimage.

Key Results¶

Generated events are preimages of numerical sets under a random variable.
Statements such as $X\le x$ are events in the original sample space.
The distribution of a random variable is defined through probabilities of generated events.

Examples¶

If $X$ is a die roll, the event generated by $X$ and the set of even numbers is rolling an even value.
If $X$ is waiting time, the statement 'wait at most five minutes' is a generated event.
If $X$ is a test score, the event 'score at least seventy' is generated by an interval of scores.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 3

Keywords: event generated by a random variable, preimage event Symbols: element of, intersection, omega, real numbers, subset

Last updated: 20 October 2026 03:02