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Bates Distribution

The probability distribution of the arithmetic mean of n independent identically distributed uniform random variables, a scaled Irwin–Hall sum.

Version
v3 · 2026-09-06 · History
Domain-specific #
1353
Origin domain
mathematics
Subdomain
probability theory
Aliases
Bates(n) distribution, Uniform-mean distribution

Core Idea

Let \(U_1,\ldots,U_n\) be independent Uniform((0,1)) random variables. Their arithmetic mean

\[ X=\frac1n\sum_{i=1}^n U_i \]

has the Bates distribution with parameter (n). Since the sum has the Irwin–Hall distribution, Bates is its scale transformation by (1/n). It is supported on ([0,1]), has mean (½), and variance (1/(12n)).[1]

The recognition invariant is fixed number (n) + iid continuous uniforms + arithmetic mean + scaled convolution law.

Structural Signature

  • Integer parameter \(n\ge1\).
  • Independent random variables.
  • Identical Uniform((0,1)) inputs, or an explicitly affinely transformed interval.
  • Arithmetic sum and division by (n).
  • Support ([0,1]).
  • Symmetry about (½).
  • Piecewise-polynomial density inherited from repeated convolution.
  • Mean (½) and variance (1/(12n)).
  • Irwin–Hall scaling relation.
  • Uniform case at (n=1).
  • Triangular density at (n=2).
  • Concentration and asymptotic normal approximation as (n) grows.
  • Parameter convention stated to prevent alternate rescalings.

What It Is Not

It is not the Irwin–Hall distribution, which describes the sum rather than the mean. It is not a mixture of uniforms; all draws contribute jointly through convolution. It is not exactly normal for finite (n), although a standardized version approaches normality.[2]

It is not the distribution of a mean from arbitrary bounded or nonidentical variables unless a generalized construction is explicitly defined.

Scope of Application

The distribution appears in simulation, business-statistics approximations, bounded random-number generation, sampling-distribution examples, and convolution exercises. Affine transformation gives the mean of iid Uniform((a,b)) variables on ([a,b]).[3]

For large (n), direct piecewise-polynomial evaluation can be numerically awkward; normal approximations or stable recurrence/convolution methods may be preferable.

Clarity

Scaling is load-bearing. If \(S=\sum U_i\), then \(S\in[0,n]\) is Irwin–Hall; \(X=S/n\in[0,1]\) is Bates. Densities transform with the Jacobian (f_X(x)=n f_S(nx)).

Independence and identical input laws are part of the named identity. Removing them changes variance and generally the entire distribution.

Manages Complexity

The family packages an (n)-fold convolution into one parameterized law. Moments, support, density shape, and limiting behavior can be reused without recomputing the distribution of each sample mean.

The compact name can obscure parameterization. Implementations should test support, scale, and moments to detect sum/mean confusion.

Abstract Reasoning

  1. Verify iid Uniform((0,1)) inputs and (n).
  2. Form the Irwin–Hall sum.
  3. Apply the (1/n) scale transformation and density Jacobian.
  4. Check support, symmetry, mean, and variance.
  5. Use exact piecewise polynomials for small/moderate (n).
  6. Quantify approximation error before using a normal limit.
  7. Apply affine transformation for Uniform((a,b)) inputs.

Knowledge Transfer

The portable structure is a sampling distribution derived by scaling a sum law. The proposed immediate parent is Probability Distribution.

Examples

n=1. The Bates distribution is Uniform((0,1)).

n=2. The mean has a symmetric triangular density on ([0,1]), peaked at (½).

Non-example. The sum of three uniforms on ([0,3]) is Irwin–Hall, not Bates until divided by three.

Structural Tensions

  • Sum versus mean parameterization.
  • Exact finite-(n) density versus normal approximation.
  • Simple moments versus piecewise-polynomial evaluation.
  • Named iid family versus generalized nonidentical inputs.
  • Bounded support versus Gaussian tail approximation.
  • Mathematical definition versus inconsistent software conventions.

Structural–Framed Character

Aggregation, scaling, convolution, support, and limiting behavior are structural. Uniform variables, densities, sample means, and distribution parameters are probability frame.

Structural Core vs. Domain Accent

The portable core is scaling an aggregate to form a reusable family. Independence, continuous uniform inputs, Irwin–Hall convolution, density pieces, and CLT approximation are constitutive domain accent.

Probability Distribution is the proposed immediate parent. Aggregation, Scaling, Convolution, Sampling, and Central Limit Theorem are related.

The prospective queue contains one strict edge to domain_specific:probability_distribution. No live DAG mutation is authorized.

Relationships to Other Abstractions

Local relationship map for Bates DistributionParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Bates DistributionDOMAINDomain-specific abstraction: Probability Distribution — is a kind ofProbabilityDistributionDOMAIN

Current abstraction Bates Distribution Domain-specific

Parents (1) — more general patterns this builds on

  • Bates Distribution is a kind of Probability Distribution Domain-specific

    Probability Distribution is the proposed immediate parent.

Hierarchy paths (5) — routes to 3 parentless roots

Neighborhood in Abstraction Space

Bates Distribution sits in a sparse region of the domain-specific corpus (96th 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

Not to Be Confused With

  • Irwin–Hall sum distribution.
  • Mixture distribution.
  • Exact Gaussian distribution.
  • Average of dependent or nonidentical variables.
  • Discrete uniform averaging.
  • A parameterization with unstated interval scaling.

References

[1] Norman L. Johnson, Samuel Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 2, 2nd ed., Wiley, 1995. registry ↩

[2] Luc Devroye, Non-Uniform Random Variate Generation, Springer, 1986. registry ↩

[3] Athanasios Papoulis and S. Unnikrishna Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed., McGraw-Hill, 2002. registry ↩

[4] William Feller, An Introduction to Probability Theory and Its Applications, vol. II, 2nd ed., Wiley, 1971. registry ↩