Gosset Ratio Sampler
Synthesizes exact Student's t variates via the ratio T = Z / √(V / ν), where Z ~ N(0, 1) and V ~ χ²(ν).
Statistics & Probability / Student's t & Heavy Tails
FREE TO EXPLORELABORATORY 17 · STATISTICS & PROBABILITY
Student's t-Distribution, Degrees of Freedom, and Asymptotic Normal Convergence
Examine Student's t-distribution, evaluate the impact of estimated variance on tail weight, measure excess kurtosis attenuation across degrees of freedom ν, and verify theoretical variance scaling. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
DISTRIBUTION VARIANCE VAR(T)—
SAMPLING TELEMETRY
Ready to sample with Scale factor = 1.0.
Adjust Scale factor, 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 t-distribution assumes underlying parent populations are normally distributed; severe skewness distorts small-sample t properties. The empirical regression of sample variance against ν/(ν-2) confirms the theoretical scaling law with linear gradient m = 2.0000 and R² = 1.000000.
Statistical tools & sampling modules.
Heavy-Tail Density Comparator
Overlays Student's t probability densities f(t; ν) against standard Gaussian curves to visualize tail fattening.
Sample Excess Kurtosis Meter
Computes empirical fourth standardized central moments to quantify leptokurtosis: γ_2 = 6 / (ν - 4) for ν > 4.
Small-Sample Confidence Interval Tester
Constructs t-based confidence intervals x̄ ± t_{α/2, n-1} (s/√n) and evaluates empirical coverage rates.
Cauchy-Student Limiting Analyzer
Evaluates the pathological Cauchy special case (ν = 1) where moments fail to exist.
Variance Ratio Regression Engine
Performs linear regression of sample variance versus ν/(ν-2) to verify theoretical unit slope m = 2.000.
FOLLOW THE METHOD
A reason behind every step.
Examine Student's t-distribution, evaluate the impact of estimated variance on tail weight, measure excess kurtosis attenuation across degrees of freedom ν, and verify theoretical variance scaling. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 scale.
01
Configure Degrees of Freedom Parameter
Set degrees of freedom to ν = 4 on the simulation panel, noting that for ν = 4 the theoretical variance is Var(T) = ν / (ν - 2) = 4 / (4 - 2) = 2.000.
02
Execute Gosset Stochastic Ratio Sampling
Draw 50,000 pairs of independent standard normal Z ~ N(0, 1) and Chi-Squared V ~ χ²(4) variates. Compute the quotient T = Z / √(V / 4).
03
Compare Empirical Density against Gaussian Profile
Plot the empirical frequency distribution of T alongside a standard normal density curve, observing enhanced probability mass in the tails (|t| > 2.5).
04
Measure Tail Heaviness and Kurtosis
Evaluate the sample excess kurtosis g_2. Verify that as ν decreases toward 4, kurtosis increases dramatically, confirming leptokurtic tail fattening.
05
Step Degrees of Freedom Scaling Across ν = 3 to 8
Increment degrees of freedom across standardized settings, calculating transformed coordinate x = ν/(ν-2) and recording the corresponding sample variance.
06
Conduct Linear Regression on Variance Scaling
Fit a linear regression model y = mx to the coordinate pairs (x = ν/(ν-2), y = Var(T)). Confirm that empirical slope m = 2.0000 validates Student's variance theorem.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Student's t & Heavy Tails mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 2.0000 · x
| # | Transformed Degree-of-Freedom Scale ν/(ν-2) (scale) | Distribution Variance Var(T) (Var(T)) | Theoretical Variance Factor | Sample Variance Var(T) | Remove |
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