Bates Distribution¶
The probability distribution of the arithmetic mean of n independent identically distributed uniform random variables, a scaled Irwin–Hall sum.
Core Idea¶
Let \(U_1,\ldots,U_n\) be independent Uniform((0,1)) random variables. Their arithmetic mean
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)).
The recognition invariant is fixed number (n) + iid continuous uniforms + arithmetic mean + scaled convolution law.
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]).
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¶
- Verify iid Uniform((0,1)) inputs and (n).
- Form the Irwin–Hall sum.
- Apply the (1/n) scale transformation and density Jacobian.
- Check support, symmetry, mean, and variance.
- Use exact piecewise polynomials for small/moderate (n).
- Quantify approximation error before using a normal limit.
- 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.
Relationships to Other Abstractions¶
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
- Bates Distribution → Probability Distribution → Random Variable → Function (Mapping)
- Bates Distribution → Probability Distribution → Probability → Measure → Set and Membership
- Bates Distribution → Probability Distribution → Probability → Measure → Aggregation → Micro Macro Linkage
- Bates Distribution → Probability Distribution → Random Variable → Probability → Measure → Set and Membership
- Bates Distribution → Probability Distribution → Random Variable → Probability → Measure → Aggregation → Micro Macro Linkage
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
- L-notation — 0.79
- Quasi-arithmetic mean — 0.76
- Box Spline — 0.76
- Delaporte Distribution — 0.76
- Cumulant — 0.75
Computed from structural-signature embeddings · 2026-09-08