CLT Zoo: the Law of Large Numbers and the Central Limit Theorem for Sixteen Distributions

Version 2 of an interactive study of sixteen distributions: Normal, Uniform, Exponential, Gamma, Beta, Lognormal, Laplace, Student-t, Pareto, Chi-square, F, Bernoulli, Binomial, Poisson, Geometric and Negative Binomial. The notebook has two views. The Distribution Explorer shows the exact density or probability function, a sample and the exact distribution function. The LLN and CLT view shows a running mean, the sampling distribution of the mean for a sample size $N$ and $M$ replications, and the standardized mean against $\mathcal N(0,1)$; the normal curves appear only when the variance is finite. The parameters, slider ranges, presets and samplers are those of version 1, so a seed gives the same draws. A short project note explains each analysis and writes out the mathematics behind it.

Contents

Path	What it is
cltzoo/distributions.py	The sixteen distributions: parameters, slider ranges and presets of version 1, samplers (identical to version 1), exact laws, exact moments, moment order, closed-form law of the sum of $N$ draws
cltzoo/clt.py	Running means, sample means, exact distance of the standardized mean to $\mathcal N(0,1)$ with and without the continuity correction, lattice Edgeworth constants
cltzoo/panels.py, cltzoo/widgets.py	The two notebook views as figure functions, and their controls
cltzoo/legacy.py	Runs the code of version 1 from legacy/clt_zoo_v1.ipynb, for comparison
notebook/clt_zoo.ipynb	The interactive notebook
tests/run_tests.py	Unit tests against version 1, numerical integration, convolution, goodness-of-fit tests and brute-force search
analysis/01_zoo.py	The zoo, its moments, and what the explorer of version 1 computed (Figure 1)
analysis/02_lln.py	The LLN on the Pareto slider: finite variance, infinite variance, infinite mean (Figure 2)
analysis/03_clt.py	The standardized mean across the zoo; exact distances for the discrete members (Figures 3, 4)
analysis/run_all.py	Runs the three analyses in order (about two minutes)
figures/	Figures as PNG (300 dpi) and PDF, every reported table as CSV
paper/zoo_note.pdf	Project note; make_paper.py writes all its numbers from figures/*.csv
legacy/	Version 1 (notebook and README), unchanged
fonts/	Latin Modern Roman (GUST Font License)

How to run

pip install -r requirements.txt
python tests/run_tests.py
cd analysis
python run_all.py
cd ../paper
python make_paper.py
xelatex zoo_note.tex && xelatex zoo_note.tex       # needs XeLaTeX

For the interactive views: pip install jupyter, then open notebook/clt_zoo.ipynb and run its cells. Seeds are fixed, so every number is the same at every run.

Usage

import numpy as np
from cltzoo import ZOO, kolmogorov_distance, corrected_error, sample_means

d = ZOO["Poisson"]
p = d.preset_params()                       # {'lambda_': 4.0}
d.moments(**p)                              # mean, var, skew, exkurt (nan where they do not exist)
d.sum_law(30, **p)                          # exact law of X_1 + ... + X_30: Poisson(120)
kolmogorov_distance(d, p, 30)               # exact sup |P(Z_30 <= x) - Phi(x)| and where it is attained
corrected_error(d, p, 30)                   # the same with the continuity correction
sample_means(d, p, N=30, M=5000, rng=np.random.default_rng(42))

Main results

• The zoo of version 1 is correct; its explorer was not. Its samplers and densities are reproduced exactly. Its distribution function was a numerical integral rescaled to end at 1 on the plotting window, with errors up to 0.022 (Pareto); Binomial and Negative Binomial stopped with an OverflowError at 76 slider positions; an impossible Uniform range ($b \le a$) was changed silently. Version 2 has none of these.
• The LLN on the Pareto slider. For $\alpha = 2.5$, 1.5 and 0.8 the median error of the running mean scales like $n^{-0.491}$, $n^{-0.332}$ and (the mean being infinite) $n^{+0.272}$, against the theoretical $n^{-1/2}$, $n^{1/\alpha - 1} = n^{-1/3}$ and $n^{+1/4}$.
• The continuity correction, measured exactly. For the five discrete members the distance of $Z_N$ to $\mathcal N(0,1)$ is $c_D/\sqrt N$ and, with the continuity correction, $c_E/\sqrt N$, with the constants of the lattice Edgeworth expansion matched to four decimals. The correction removes the lattice term and leaves the skewness term: to bring the error below 0.01, Bernoulli(0.5) needs $N = 1{,}592$ without it and $N = 4$ with it; Geometric(0.3) needs 427 and 183.

All numbers are in figures/*.csv and in the project note.

Changes from version 1

See NEWS.md: what was added and a table of the corrections to version 1.

Citation

See CITATION.cff, or:

Yılmaz, M. İ. (2026). CLT Zoo: the Law of Large Numbers and the Central Limit Theorem for sixteen distributions (Version 2.1.0) [Computer software]. https://github.com/mikbalyilmaz/CLT-Zoo-exercise

License

MIT, see LICENSE.