Non-Gaussian Parent Variate Generator
Generates independent random samples from Uniform U[0, 1], Exponential(1), and Bernoulli(0.2) parent distributions.
Statistics & Probability / Central Limit Theorem
FREE TO EXPLORELABORATORY 07 · STATISTICS & PROBABILITY
Central Limit Theorem and Lindeberg-Lévy Asymptotics
Demonstrate the asymptotic normality of standardized sample sums from non-Gaussian parent distributions, verify the Berry-Esseen theorem error bounds, and quantify standard error scaling SE = σ / √N. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
STANDARD ERROR SE—
SAMPLING TELEMETRY
Ready to sample with Sample Size N = 10.0.
Adjust Sample Size N, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
The classical CLT applies only to sums of variables with finite second moments; it fails for heavy-tailed distributions like Cauchy or Pareto (α ≤ 2). The empirical regression of ln(SE) versus ln(N) yields an empirical slope of m = -0.5000, confirming the theoretical CLT root-N rate with R² = 1.000000.
Statistical tools & sampling modules.
Sample Mean Standardizer
Computes standardized sums Z_N = (X̄_N - μ) / (σ / √N) to map empirical distributions onto the standard normal domain.
Gaussian Normality Comparator
Evaluates the Kolmogorov supremum distance sup_z |F_N(z) - Φ(z)| against the standard normal cumulative distribution function.
Berry-Esseen Bound Evaluator
Calculates the theoretical Berry-Esseen upper bound C · ρ / (σ³ √N) where ρ = E[|X - μ|³] is the third absolute central moment.
Quantile-Quantile (Q-Q) Plot Engine
Plots empirical standardized quantiles against theoretical standard normal quantiles to visually audit tail normality.
Root-N Regression Compiler
Performs linear regression on ln(SE) versus ln(N) to verify the theoretical -0.500 (-1/2) square-root scaling exponent.
FOLLOW THE METHOD
A reason behind every step.
Demonstrate the asymptotic normality of standardized sample sums from non-Gaussian parent distributions, verify the Berry-Esseen theorem error bounds, and quantify standard error scaling SE = σ / √N. Standardized parameter settings: 10.0, 20.0, 40.0, 80.0, 160.0, 320.0 samples.
01
Configure Non-Gaussian Parent Distribution
Select the continuous uniform distribution U[0, 1] with known mean μ = 0.5 and variance σ² = 1/12 ≈ 0.08333 (σ = 0.28868).
02
Simulate Standardized Sample Averages for N = 10
Set sample size N = 10. Draw 5,000 independent samples of size N, evaluate the sample mean X̄ for each, and compute standardized scores Z = (X̄ - μ) / (σ / √N).
03
Measure Empirical Standard Error
Calculate the standard deviation of the 5,000 sample means to obtain the empirical standard error SE = σ / √N. For N = 10, SE = 0.28868 / √10 = 0.09129.
04
Evaluate Berry-Esseen Normality Convergence
Measure the Kolmogorov distance between the empirical distribution of Z and the Gaussian CDF Φ(z), verifying that the deviation satisfies the Berry-Esseen bound.
05
Step Sample Size Across N = 10 to 320
Increment sample size across N ∈ {10, 20, 40, 80, 160, 320}, observing rapid convergence of the sampling distribution to the bell-shaped Gaussian curve.
06
Execute Log-Log Regression on Standard Error
Fit a linear regression model to the coordinates (x = ln(N), y = ln(SE)). Confirm that the empirical gradient m = -0.5000 rigorously verifies the 1/√N standard error scaling.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Central Limit Theorem mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
ln(SE) = -0.5000 · ln(N) + 1.0000
| # | Sample Size N (samples) | Standard Error SE (SE) | Natural Log of Sample Size ln(N) | Natural Log of Standard Error ln(SE) | Remove |
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