Random Interval¶

Definition¶

Let $A$ and $B$ be random variables with $A\le B$ $B$ . A random interval is an interval whose endpoints are random, such as $[A,B]$ .

Interpretation¶

A random interval is an interval whose position and length depend on the outcome of the experiment. Before the experiment is run, $[A,B]$ is unknown; afterwards it is an ordinary interval of numbers.

Six random intervals from repeated runs drawn as horizontal segments above a fixed vertical line at theta; five of them cover theta and one misses it.

Each horizontal segment is the interval $[A,B]$ obtained in one run, and the dashed vertical line is a fixed number $\theta$ that is the same in every run. Five of the six segments contain $\theta$ ; the orange one lies entirely to its right. The randomness sits in the endpoints, not in $\theta$ .

The basic event is that the interval covers $\theta$ :

\{A\le\theta\le B\}=\{A\le\theta\}\cap\{B\ge\theta\}.

Its probability $\mathbb{P}(A\le\theta\le B)$ is the coverage probability. Because $A\le B$ $B$ , the interval can miss $\theta$ in two disjoint ways, lying entirely to the right ( $A >$ $A>\theta$ $θ$ ) or entirely to the left ( $B <$ $B<\theta$ $θ$ ), so the complement rule gives

\mathbb{P}(A\le\theta\le B) =1-\mathbb{P}(A>\theta)-\mathbb{P}(B<\theta).

The fraction $5/6$ in the figure is what happened in six runs, not the coverage probability itself.

The probability statement is about the procedure before the run. After one run, $[A,B]$ is a fixed interval that either contains the fixed $\theta$ or does not. A natural example is the interval $[X_{(1)},X_{(n)}]$ between the smallest and the largest order statistic of a sample. The idea returns in statistics as the confidence interval, a random interval constructed so that its coverage probability takes a chosen value such as $0.95$ .

Key Results¶

A random interval has endpoints that are random variables.
Events can be formed from statements about whether a fixed value lies inside the interval.
Random intervals are useful for uncertainty bands and coverage statements.

Examples¶

A confidence interval computed from data is a random interval before the data are observed.
The interval between two random arrival times is random.
A forecast range with random lower and upper bounds is a random interval.

Prerequisites¶

Random Variable

Sources¶

Probability Theory for EOR · Week 5

Keywords: interval with random endpoints Symbols: intersection, theta

Last updated: 22 October 2026 19:04