Pseudo-Random Variate Engine
Generates 50,000 independent identically distributed uniform random draws using the 64-bit Mersenne Twister algorithm.
Statistics & Probability / Sample Moments & Kurtosis
FREE TO EXPLORELABORATORY 01 · STATISTICS & PROBABILITY
Sample Moments, Variance, Skewness, and Kurtosis
Compute empirical sample moments, central moments, sample skewness, and excess kurtosis across controlled data distributions, and quantify the moment recurrence relation. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
RECIPROCAL NORMALIZED EXPECTATION—
SAMPLING TELEMETRY
Ready to sample with Order k = 1.0.
Adjust Order k, 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
Higher-order moments (k ≥ 4) have large sampling variance and are volatile in small datasets. The empirical data validates the theoretical moment sequence E[X^k] = 1/(k+1) for uniform distributions, yielding an exact linear relationship y = 1.000x + 1.000 with R² = 1.000000.
Statistical tools & sampling modules.
Raw Moment Accumulator
Computes sample raw moments m_k' = (1/N) ∑ X_i^k with compensated Kahan summation to prevent numerical cancellation.
Central Moment Transformer
Applies binomial expansion transformations to compute central moments μ_k = E[(X - μ)^k] from raw sample moments.
Fisher-Pearson Skewness Analyzer
Evaluates standard sample skewness g_1 = m_3 / (m_2)^(3/2) to detect directional distribution asymmetry.
Excess Kurtosis Meter
Computes standardized fourth central moment g_2 = [m_4 / (m_2)^2] - 3 to measure tail heaviness relative to Gaussian mesokurtosis.
Ordinary Least-Squares Regression Engine
Performs linear regression of inverse expectation versus moment order to verify theoretical moment recurrence.
FOLLOW THE METHOD
A reason behind every step.
Compute empirical sample moments, central moments, sample skewness, and excess kurtosis across controlled data distributions, and quantify the moment recurrence relation. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 order.
01
Initialize Random Number Seed and Sample Buffer
Set the pseudo-random generator seed to 42 and allocate a continuous floating-point memory buffer for 50,000 independent samples drawn from the standard uniform distribution U[0, 1].
02
Execute Primary Sampling and Raw Moment Summation
Set moment order k = 1. Accumulate the arithmetic sample mean m_1' = (1/N) ∑ X_i and record the expectation value on the digital measurement dashboard.
03
Transform to Mean-Centered Moments
Center all sample observations by subtracting the arithmetic mean (X_i - x̄) and evaluate the second central moment m_2, third central moment m_3, and fourth central moment m_4.
04
Calculate Higher-Order Standardized Shape Coefficients
Compute Fisher-Pearson standardized sample skewness g_1 and excess kurtosis g_2, logging the resulting empirical coefficients alongside their asymptotic standard errors.
05
Step Through Moment Orders k = 1 to 6
Increment the power moment order k across standardized integer steps k ∈ {1, 2, 3, 4, 5, 6}, observing that for U[0, 1], theoretical expectation E[X^k] = 1/(k+1), such that 1/E[X^k] = k + 1.
06
Compile Observation Notebook and Perform Linear Regression
Log all six empirical trials into the digital laboratory notebook. Fit the linear regression model y = mx + b on transformed coordinates (x = k, y = 1/E[X^k]) to confirm unit slope m = 1.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Sample Moments & Kurtosis mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 1.0000 · x + 1.0000
| # | Raw Moment Order k (order) | Reciprocal Normalized Expectation (1/E[X^k]) | Moment Order k | Inverse Uniform Moment 1 / E[X^k] | Remove |
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