Gamma Random Variate Generator
Generates independent Gamma(α, θ) random variates using Marsaglia and Tsang's fast rejection method.
Statistics & Probability / Gamma & Chi-Squared Family
FREE TO EXPLORELABORATORY 16 · STATISTICS & PROBABILITY
Gamma Distribution Family, Erlang Queuing Times, and Chi-Squared Connections
Investigate the continuous Gamma distribution family, demonstrate Erlang waiting times as sums of independent exponential variates, establish the Chi-Squared distribution as a specialized Gamma model, and verify degrees-of-freedom variance scaling. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
THEORETICAL DISTRIBUTION VARIANCE VAR(X)—
SAMPLING TELEMETRY
Ready to sample with DoF k = 1.0.
Adjust DoF 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
The Gamma distribution is restricted strictly to non-negative real support (x > 0) and cannot model negative observations. The empirical regression of sample variance against degrees of freedom k confirms the theoretical scaling Var(X) = 2k with slope m = 2.0000 and R² = 1.000000.
Statistical tools & sampling modules.
Erlang Convolution Accumulator
Simulates the sum of k independent Exponential(λ) waiting times to construct Erlang-k distributions.
Sum-of-Squared-Normals Engine
Squares and aggregates k independent standard Gaussian variates to generate Chi-Squared χ²(k) variates.
Gamma Shape-Scale Parameter Resolver
Computes empirical shape α̂ and scale θ̂ estimators via maximum likelihood and method of moments.
Chi-Squared Quantile Comparator
Compares empirical distribution percentiles against theoretical χ² critical values across multiple degrees of freedom.
Degrees-of-Freedom Variance Regression Unit
Performs linear regression of variance versus degrees of freedom k to confirm the theoretical factor m = 2.000.
FOLLOW THE METHOD
A reason behind every step.
Investigate the continuous Gamma distribution family, demonstrate Erlang waiting times as sums of independent exponential variates, establish the Chi-Squared distribution as a specialized Gamma model, and verify degrees-of-freedom variance scaling. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 dof.
01
Configure Degrees of Freedom Parameter
Set degrees of freedom k = 1 on the control interface, initializing the generator to evaluate the base Chi-Squared distribution χ²(1).
02
Synthesize Chi-Squared Variates from Gaussian Draws
Draw 50,000 standard normal variates Z_i ~ N(0, 1), square each variate X = Z_i^2, and verify that the resulting empirical distribution matches the Gamma(1/2, 2) density.
03
Simulate Erlang Queuing Times via Exponential Sums
Set integer shape k = 3 and sum three independent Exponential(1.0) arrival intervals. Verify that the convolution reproduces the Erlang-3 distribution.
04
Track Mean and Variance Across Degrees of Freedom
Record the empirical mean E[X] = k and sample variance Var(X) = 2k on the digital laboratory notebook dashboard.
05
Step Degrees of Freedom Parameter Across k = 1 to 6
Increment the degrees of freedom parameter across standardized integer steps k ∈ {1, 2, 3, 4, 5, 6}, logging the corresponding sample variances.
06
Conduct Linear Regression on Variance Scaling
Fit a linear regression model y = mx to the coordinate pairs (x = k, y = Var(X)). Confirm that empirical slope m = 2.0000 validates the fundamental scaling law Var(χ²(k)) = 2k.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Gamma & Chi-Squared Family mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 2.0000 · x
| # | Degrees of Freedom Parameter k (dof) | Theoretical Distribution Variance Var(X) (Var(X)) | Degrees of Freedom k | Distribution Variance Var(X) | Remove |
|---|
No readings yet. Execute a statistical trial and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING