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Uniform discrete distribution

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The Discrete Uniform distribution assigns equal probability to every integer in an inclusive interval. It is the right model when an integer is selected without preference between min and max.

Feature Reading
Support Support is {min, min+1,…,max}. Unlike continuous Uniform, each point has positive mass and the number of outcomes is finite.
Shape It is Uniform’s discrete counterpart, but not an automatic discretization: rounding a continuous uniform does not generally yield equal masses. It also underlies simple unweighted sampling.

Origin and terminology

Discrete equiprobability appears in games of chance, urns, and random-integer generation. Its value is making a finite outcome space explicit.

In real models

  • random selection of integers, cards, or indices
  • simulation of ordered categories with equal probability

Comparisons

Support is inclusive: changing max to max-1 changes outcome count and every moment.

A fair die needs exact endpoints

The Discrete Uniform distribution is the law of an ideal die whose faces carry consecutive integer labels. Both endpoints are included, so the number of possible outcomes is max-min+1. Forgetting that extra one is a common source of errors in simulation and array indexing.

Equal probability for every integer is a strong claim about the mechanism, not an automatic expression of ignorance. A random number generator may also introduce modulo bias when its native range is not a multiple of the desired number of outcomes. Beyond simulation, the model suits well-designed lotteries and randomly selected ordered categories. It does not usually fit rounded continuous measurements, whose endpoint masses follow a different rule.

Decision guide

A good candidate when: only consecutive integers between known bounds are possible and all are genuinely equally likely.

Compare it with: Categorical for unequal probabilities and continuous Uniform for nondiscrete measurements. Rounding a continuous Uniform variable does not always make endpoint integers equally likely.

References

  • SciPy reference: scipy.stats.randint — definition and parameterization
  • Johnson, N. L., Kemp, A. W. & Kotz, S. (2005). Univariate Discrete Distributions, 3rd ed. Wiley.
  • Evans, M., Hastings, N. & Peacock, B. (2000). Statistical Distributions, 3rd ed. Wiley.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley.

Uniform Distribution: equations and calculator

Distribution defintion

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

Distribution domain

x∈{a,a+1,…,b−1,b}x\in \{a,a+1,\dots,b-1,b\}

Parameters domain and parameters constraints

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

Cumulative distribution function

FX(x)=x−a+1b−a+1F_{X}\left(x\right)=\frac{x -a+1}{b-a+1}

Probability mass function

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

Percent point function/Sample

FX−1(u)=⌈u(b−a+1)+a−1⌉F^{-1}_{X}\left(u\right)=\left\lceil u(b-a+1)+a-1 \right\rceil

Parametric centered moments

E[Xk]=μk′=∑x=abxkfX(x)=1b−a+1∑x=abxkE[X^k]=\mu'_{k}=\sum_{x=a}^{b}x^{k}f_{X}\left(x\right)=\frac{1}{b-a+1}\sum_{x=a}^{b}x^{k}

Parametric mean

Mean(X)=μ1′=a+b2\mathrm{Mean}(X)=\mu'_{1}=\frac{a+b}{2}

Parametric variance

Variance(X)=(μ2′−μ1′2)=(b−a+1)2−112\mathrm{Variance}(X)=(\mu'_{2}-\mu'^{2}_{1})=\frac{(b-a+1)^2-1}{12}

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=3−6((b−a+1)2+1)5((b−a+1)2−1)\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((b-a+1)^2+1)}{5((b-a+1)^2-1)}

Parametric median

Median(X)=a+b2\mathrm{Median}(X)=\frac{a+b}{2}

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}
⌈x⌉:Ceiling Function\lceil{x}\rceil: \text{Ceiling Function}