PLAYGROUND

Bates distribution

From phenomenon to model

It is named after Grace E. Bates and is closely related to Irwin–Hall, which describes the sum rather than the average of Uniform variables. Keeping the scaling straight prevents confusing the two families.

What to look at

The support remains [min,max], but its effective spread shrinks like 1/√n. At n=1 it is uniform; for large n, the central limit theorem explains its nearly Gaussian shape.

Case Interpretation
n = 1 uniform over the interval
increasing n more concentration at the centre
n → ∞ normal approximation after standardization

Names you may also encounter

It is also called the Bates distribution; the sum version is Irwin–Hall. Their piecewise-polynomial densities look related, but their scales and means are different.

When it makes sense

It can represent an average of independent bounded contributions, such as a composite time with many small uniform perturbations. It is not a first choice when the summands have tails, dependence, or strongly unequal variability.

What not to assume

The n parameter must be a positive integer. A fractional value may produce a plausible-looking curve, but it no longer represents this averaging mechanism.

Grace Bates and the average of several uniform chances

The distribution honours American mathematician Grace E. Bates. It is obtained by averaging n independent Uniform variables. Their sum, before division by n, follows the Irwin Hall distribution. At n=1 it returns to Uniform, and at n=2 it becomes a symmetric Triangular law.

As n increases, the central limit theorem rounds the curve and reduces its variance, while support remains bounded. This combination can simulate an average of small disturbances that never leaves known limits. The model is less compelling when n is treated merely as a smoothness control: its natural meaning is an integer count of independent contributions on a common scale.

Decision guide

A good candidate when: the observation is the average of a fixed number of independent uniform contributions with common bounds.

Compare it with: Triangular, Irwin–Hall, and Normal. Bates models an average whereas Irwin–Hall models the sum; an approximately bell-shaped sample does not prove uniform components.

References

  • Wolfram Language reference: BatesDistribution — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley.
  • Irwin, J. O. (1955). A unified derivation of some well-known frequency distributions of interest in biometry and statistics. Journal of the Royal Statistical Society, Series A, 118(4), 389–396.

Bates Distribution: equations and calculator

Distribution defintion

X∼Bates(n,min,max)X\sim\mathrm{Bates}\left(n,\text{min},\text{max}\right)

Distribution domain

x∈(min,max)x\in\left(\text{min},\text{max}\right)

Parameters domain and parameters constraints

n∈N+,min∈R,max∈Rn\in\mathbb{N}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R}

Cumulative distribution function

FX(x)=1n!∑k=0⌊y⌋(−1)k(nk)(y−k)nF_{X}\left(x\right)=\frac{1}{n!}\sum_{k=0}^{\lfloor y\rfloor}\left(-1\right)^{k}\binom{n}{k}\left(y-k\right)^{n}

Probability density function

fX(x)=n(max−min)(n−1)!∑k=0⌊y⌋(−1)k(nk)(y−k)n−1f_{X}\left(x\right)=\frac{n}{\left(\text{max}-\text{min}\right)(n-1)!}\sum_{k=0}^{\lfloor y\rfloor}\left(-1\right)^{k}\binom{n}{k}\left(y-k\right)^{n-1}

Percent point function/Sample

FX−1(u)=numerical inversion of FXF^{-1}_{X}\left(u\right)=\text{numerical inversion of }F_{X}

Non-central parametric moments

μ~k=E[(Y−12)k]\tilde{\mu}_{k}=E\left[\left(Y-\tfrac{1}{2}\right)^{k}\right]

Parametric mean

Mean(X)=min+max2\mathrm{Mean}(X)=\frac{\text{min}+\text{max}}{2}

Parametric variance

Variance(X)=(max−min)212n\mathrm{Variance}(X)=\frac{\left(\text{max}-\text{min}\right)^{2}}{12n}

Parametric skewness

Skewness(X)=0\mathrm{Skewness}(X)=0

Parametric kurtosis

Kurtosis(X)=3−65n\mathrm{Kurtosis}(X)=3-\frac{6}{5n}

Parametric median

Median(X)=min+max2\mathrm{Median}(X)=\frac{\text{min}+\text{max}}{2}

Parametric mode

Mode(X)=min+max2\mathrm{Mode}(X)=\frac{\text{min}+\text{max}}{2}

Additional information and definitions

Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution.
X=min+(max−min)⋅1n∑i=1nUiX=\text{min}+\left(\text{max}-\text{min}\right)\cdot\frac{1}{n}\sum_{i=1}^{n}U_{i}
y=n⋅(x−min)/(max−min)y=n\cdot\left(x-\text{min}\right)/\left(\text{max}-\text{min}\right)
n:Number of averaged uniforms (Irwin-Hall order)n:\text{Number of averaged uniforms (Irwin-Hall order)}
min:Lower bound of the support\text{min}:\text{Lower bound of the support}
max:Upper bound of the support\text{max}:\text{Upper bound of the support}
(nk):Binomial coefficient\binom{n}{k}:\text{Binomial coefficient}
Ui:Uniform[0,1] random varibleU_{i}:\text{Uniform[0,1] random varible}