Continuous Uniform Distribution¶

The bounded continuous probability law whose density is constant, assigning probability in direct proportion to interval length.

Version: v2 · 2026-09-06 · History
Domain-specific #: 1554
Origin domain: probability
Subdomain: continuous distributions
Aliases: Rectangular Distribution, Uniform Distribution on an Interval

Core Idea¶

The continuous uniform distribution on a finite interval assigns the same probability density to every point of that interval. If \(a<b\), a random variable \(X\sim U(a,b)\) has density

\[ f_X(x)= \begin{cases} \dfrac{1}{b-a}, & a\le x\le b,\\ 0, & \text{otherwise}. \end{cases} \]

Consequently, for any subinterval \([c,d]\subseteq[a,b]\),

\[ \Pr(c\le X\le d)=\frac{d-c}{b-a}. \]

Equal length, not equal listed points, is the invariant. Each individual point still has probability zero. NIST describes the law as equal probability over a given continuous range and identifies \(U(0,1)\) as its standard form.

Scope of Application¶

Continuous uniform laws model quantities known only to lie in a bounded interval when equal-length regions are assigned equal probability. Examples include an idealized random phase on a cycle represented over one period, randomized start times within a window, simulation inputs sampled over fixed limits, and rounding error under specific phase assumptions.

The standard uniform distribution is central to simulation. Pseudorandom generators typically expose values approximating \(U(0,1)\); transformations then generate other laws. NIST highlights random-number generation as a major application of the standard form.

Clarity¶

“Every value is equally likely” is useful intuition but mathematically misleading for a continuum. Any exact value has probability zero. The precise statement is that equal-length measurable subsets receive equal probability, and probability is the integral of a constant density.

Endpoint inclusion does not change the distribution: \([a,b]\), \((a,b)\), and half-open variants differ only by sets of probability zero.

Manages Complexity¶

The law reduces an uncertain bounded quantity to two parameters. Once \(a\) and \(b\) are fixed, normalization, interval probabilities, quantiles, moments, and sampling all follow immediately. NIST gives mean \((a+b)/2\) and standard deviation \((b-a)/\sqrt{12}\).

The quantile function is especially simple:

\[ F^{-1}(p)=a+p(b-a),\qquad 0\le p\le1. \]

Abstract Reasoning¶

Normalization follows because the rectangular area is \((b-a)\times1/(b-a)=1\). Integrating \(x/(b-a)\) over the interval yields

\[ \mathbb E[X]=\frac{a+b}{2}, \qquad \operatorname{Var}(X)=\frac{(b-a)^2}{12}. \]

For \(U\sim U(0,1)\), the event \(a+(b-a)U\le x\) is equivalent to \(U\le(x-a)/(b-a)\), giving the piecewise-linear cumulative distribution. Conversely, standardization \((X-a)/(b-a)\) produces \(U(0,1)\).

Knowledge Transfer¶

Literal transfer occurs whenever the same bounded interval, constant density, and length measure are present. Simulation, reliability windows, randomized algorithms, and simplified physical phase models reuse the exact equations.

Transfer to a disk or sphere requires replacing interval length with normalized area or surface measure. The higher-level pattern—constant density relative to a chosen base measure—survives, but the one-dimensional formulas do not. Transfer to discrete cases replaces density with point mass and therefore changes the identity.

Relationships to Other Abstractions¶

Current abstraction Continuous Uniform Distribution Domain-specific

Parents (1) — more general patterns this builds on

Continuous Uniform Distribution is a kind of Probability Distribution Domain-specific

domain_specific:probability_distribution is the minimal parent because \(U(a,b)\) is a particular continuous probability law.

Neighborhood in Abstraction Space¶

Continuous Uniform Distribution sits in a sparse region of the domain-specific corpus (86^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08