Categorical Random Variate Generator
Executes single-trial categorical draws across K discrete outcomes using cumulative probability interval mapping.
Statistics & Probability / Multinomial Joint Sampling
FREE TO EXPLORELABORATORY 21 · STATISTICS & PROBABILITY
Multinomial Distribution, Categorical Probabilities, and Covariance Structure
Investigate the multivariate Multinomial distribution, simulate multi-category discrete trials, evaluate joint probability mass functions, quantify the negative covariance between competing categories, and extract the pairwise covariance coefficient. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
ABSOLUTE COVARIANCE |COV(X_1, X_2)|—
SAMPLING TELEMETRY
Ready to sample with Trials Factor k = 1.0.
Adjust Trials Factor 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 standard Multinomial model assumes constant category probabilities, which fails in urn models without replacement. The empirical regression of absolute covariance against trial multiplier k confirms the theoretical parameter scaling with slope m = 0.0600 and R² = 1.000000.
Statistical tools & sampling modules.
Multivariate Count Accumulator
Logs simultaneous category tallies X = (X_1, X_2, ..., X_K) across n independent multinomial trials.
Joint PMF Analytical Evaluator
Computes exact multinomial probabilities P(X = x) = (n! / ∏ x_i!) ∏ p_i^(x_i) via logarithmic factorial sums.
Empirical Covariance Matrix Resolver
Computes the full K × K sample covariance matrix, verifying negative off-diagonal entries Cov(X_i, X_j) = -n p_i p_j.
Marginal Binomial Invariance Tester
Verifies that the marginal distribution of any single category X_i is exactly Binomial(n, p_i).
Linear Covariance Regression Engine
Performs linear regression of |Cov(X_1, X_2)| versus trial factor to extract the theoretical product p_1 p_2 = 0.060.
FOLLOW THE METHOD
A reason behind every step.
Investigate the multivariate Multinomial distribution, simulate multi-category discrete trials, evaluate joint probability mass functions, quantify the negative covariance between competing categories, and extract the pairwise covariance coefficient. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 factor.
01
Configure Categorical Probability Vector
Set outcome categories to K = 3 with probability vector p = [0.20, 0.30, 0.50] on the simulation dashboard, noting that ∑ p_i = 1.00.
02
Execute Primary Multinomial Batch (n = 10 trials)
Run an ensemble of 20,000 multinomial experiments with trial count n = 10 · k for k = 1.0. Record the tripartite count vectors (X_1, X_2, X_3).
03
Verify Marginal Binomial Behavior
Extract counts for category 1 (X_1) and category 2 (X_2). Verify that marginal expectations match E[X_1] = 10 · 0.20 = 2.0 and E[X_2] = 10 · 0.30 = 3.0.
04
Evaluate Cross-Category Covariance
Compute empirical sample covariance Cov(X_1, X_2) = E[X_1 X_2] - E[X_1]E[X_2]. Confirm that the result is strictly negative: Cov(X_1, X_2) = -10 · 0.20 · 0.30 = -0.60.
05
Step Trial Factor Across k = 1 to 6
Increment the trial factor across standardized settings k ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, measuring the absolute covariance magnitude |Cov(X_1, X_2)| = 0.060 · (10k).
06
Perform Linear Regression on Covariance Scaling
Fit a linear regression model y = mx to the coordinate pairs (x = k, y = |Cov(X_1, X_2)| / 10). Confirm that empirical slope m = 0.0600 recovers the exact probability product p_1 p_2.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Multinomial Joint Sampling mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.0600 · x
| # | Total Trial Multiplier Factor k (factor) | Absolute Covariance |Cov(X_1, X_2)| (|Cov|) | Trial Factor k | Absolute Covariance |Cov(X_1, X_2)| | Remove |
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