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Uniform distribution

What it represents

The Uniform distribution assigns equal density to every value in an interval. It is the maximum-symmetry model when only two bounds are known and no part of the support is preferred.

Historical clue

Uniform is a basic probability construction and the standard source for pseudo-random numbers. Its simplicity makes it foundational for simulation and transformations.

Relationships that clarify its use

Uniform transforms generate many laws: powers for Power Function, squared sine for Arcsine, and averages for Bates. The discrete version assigns mass rather than density to integers.

Data examples

  • random sampling within known bounds
  • interval uncertainty, simulation, and quantile-based generation of other laws

Modelling warning

Do not equate it with “unbiased” on every scale; state which measurement should be flat.

The rule that turns an interval into a laboratory

A Uniform variable between zero and one is the raw material of inverse transform simulation. If F is a continuous distribution function, applying its quantile function to an ideal uniform variable generates an observation distributed as F. Many Monte Carlo routines begin with this idea, even when they later need faster or more stable algorithms.

Among all continuous distributions confined to fixed endpoints, Uniform has maximum entropy. This property formalizes a very specific absence of preference within the interval. It does not justify the model when all we have are uncertain minimum and maximum guesses, or when the endpoints are sample extremes. In those settings, apparent neutrality can conceal an excessively rigid claim.

Decision guide

A good candidate when: firm bounds exist and no information makes one part of the interval more plausible than another.

Compare it with: Triangular, PERT, and Beta when central tendency or shape information exists. Estimating bounds only from observed minimum and maximum usually understates out-of-sample uncertainty.

References

Uniform Distribution: equations and calculator

Distribution defintion

X∼Uniform(a,b)X\sim\mathrm{Uniform}\left(a,b\right)

Distribution domain

x∈[a,b]x\in [a,b]

Parameters domain and parameters constraints

a∈R,b∈R,a<ba\in\mathbb{R},b\in\mathbb{R},a < b

Cumulative distribution function

FX(x)=x−ab−aF_{X}\left(x\right)=\frac{x-a}{b-a}

Probability density function

fX(x)=1b−af_{X}\left(x\right)=\frac{1}{b-a}

Percent point function/Sample

FX−1(u)=a+u⋅(b−a)F^{-1}_{X}\left(u\right)=a+u\cdot(b-a)

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx=1k+1∑i=0kaibk−i\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx=\frac{1}{k+1}\sum_{i=0}^k a^ib^{k-i}

Parametric mean

Mean(X)=μ1′=12(a+b)\mathrm{Mean}(X)=\mu'_{1}=\tfrac{1}{2}(a+b)

Parametric variance

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=0\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=0

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3−65\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=3-\frac{6}{5}

Parametric median

Median(X)=12(a+b)\mathrm{Median}(X)=\tfrac{1}{2}(a+b)

Parametric mode

Mode(X)∈[a,b]\mathrm{Mode}(X)\in [a,b]

Additional information and definitions

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}