Continuous Uniform Distribution¶

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

Version: v3 · 2026-09-19 · History
Domain-specific #: 1554
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Probability, Continuous Distributions → Mathematics
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.

How would you explain it like I'm…

No-Favorite-Spot Stick

Imagine a stick, and a ladybug lands somewhere on it without any favorite spot. Any two pieces of the stick that are the same length are equally likely to get the ladybug. A bigger piece gets a bigger chance, just because it is longer.

Equal Lengths, Equal Chances

Suppose a number is picked from between 0 and 10, and no part of that range is favored. That is a continuous uniform distribution. The chance of landing in some stretch depends only on how long the stretch is: landing between 2 and 4 is just as likely as between 7 and 9, and landing between 0 and 5 has a one-half chance. Because there are endlessly many possible numbers, the chance of hitting exactly one particular number, like exactly 3, is zero. What is 'equal' is chances for equal lengths.

Constant-Density Interval Distribution

The continuous uniform distribution on an interval from a to b (with a < b) gives every point in the interval the same probability density, 1/(b - a), and density zero outside. The probability of landing in a subinterval from c to d is its length divided by the whole length: (d - c)/(b - a). So equal lengths get equal probability. It is easy to misread 'same density' as 'every point is equally likely': in fact each single point has probability zero, and only intervals get positive probability. The standard version is U(0, 1), the uniform distribution on 0 to 1, and it is a specific family of probability distributions, not a general idea of fairness.

The continuous uniform distribution U(a, b), for real a < b, has density f(x) = 1/(b - a) for a <= x <= b and 0 otherwise. For any subinterval [c, d] of [a, b], P(c <= X <= d) = (d - c)/(b - a), so probability is proportional to length: equal length, not equal listed points, is the invariant. Every individual point has probability zero, even though the density is constant on the support. The standard form is U(0, 1), and other uniform laws are obtained by affine transformation. The family is a specific probability distribution, not the general notion of fairness or sameness. It recurs across probability, simulation, measurement-error models, and reference calculations because of its simple support, constant density, affine behaviour, easily computed moments, direct sampling method, and hard boundaries at a and b.

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) — outgoing typed relations

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

domain_specific:probability_distribution is the broader abstraction this entry instantiates because \(U(a,b)\) is a particular continuous probability law.

Hierarchy paths (3) — routes to 2 parentless roots

Continuous Uniform Distribution → is a kind of Probability Distribution → presupposes, typical Random Variable → is a kind of Function (Mapping)

Show alternative paths (2)

Neighborhood in Abstraction Space¶

Continuous Uniform Distribution sits in a sparse region of the domain-specific corpus (89^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Numerical Root-Finding & Quadrature Methods (8 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25