Continuous Uniform Distribution

Compact study note.

Summary

The continuous uniform distribution gives constant density over a finite interval. It models an ideal measurement equally likely across that interval.^[1]

Prerequisites

Continuous Random Variable

Definition

Distribution notation:

X \sim Uniform (l, u), l < u .

Notation and Assumptions

Every subinterval probability is proportional to its length.

Parameters

l, u \in R, l < u .

Support

$[a, b]$ ; endpoint choice does not change probabilities.

PMF or PDF

f_{X} (x) = 1 / (u - l), l \leq x \leq u,

and $0$ otherwise.

CDF

F_{X} (x) = 0, x < l .

F_{X} (x) = \frac{x - l}{u - l}, l \leq x \leq u .

F_{X} (x) = 1, x > u .

Moments

E [X] = (l + u) / 2.

Var (X) = (u - l)^{2} / 12.

M_{X} (t) = \frac{\exp (t u) - \exp (t l)}{t (u - l)}, t \neq 0.

Essential Result

Probabilities are interval lengths divided by total length.

Small Example

If $X \sim Uniform (0, 30)$ , then $P (5 < X < 15) = 10 / 30 = 1 / 3$ .

Common Mistakes

Saying each individual real value is equally probable with positive probability.
Forgetting the density changes when interval length changes.

Connections

References

OpenStax, Introductory Statistics 2e, "Chapter 5: Continuous Random Variables", https://openstax.org/books/introductory-statistics-2e/pages/5-introduction ↩︎