Bernoulli Trial Pseudo-Generator
Executes independent binary stochastic trials with user-calibrated success probability p using uniform thresholding.
Statistics & Probability / Binomial & Hypergeometric Sampling
FREE TO EXPLORELABORATORY 12 · STATISTICS & PROBABILITY
Bernoulli Process, Binomial Distribution, and Hypergeometric Sampling
Investigate discrete counting processes, evaluate Bernoulli trials, contrast Binomial sampling with replacement against Hypergeometric sampling without replacement, and extract the single-trial Bernoulli variance. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
SCALED BINOMIAL VARIANCE VAR(X)/N—
SAMPLING TELEMETRY
Ready to sample with Factor k = 1.0.
Adjust 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
Binomial models assume strict outcome independence, which fails in contagious or clustered event series. The empirical regression of cumulative variance against trial factor k confirms the theoretical single-trial variance m = 0.2400 with R² = 1.000000.
Statistical tools & sampling modules.
Combinatorial Binomial Engine
Calculates exact binomial probabilities P(X = k) = (n choose k) p^k (1-p)^(n-k) via logarithmic gamma recursions.
Urn Finite Population Simulator
Simulates sampling without replacement from finite urn structures to model hypergeometric probabilities.
Sampling Variance Tracker
Computes running sample variances across trial batches to quantify the variance reduction factor (N - n)/(N - 1).
Poisson Limit Comparator
Overlays Poisson approximations when n is large and p is small to measure total variation distance.
Linear Variance Regression Unit
Fits linear models to observed variance scaling to extract the fundamental Bernoulli variance parameter p(1-p).
FOLLOW THE METHOD
A reason behind every step.
Investigate discrete counting processes, evaluate Bernoulli trials, contrast Binomial sampling with replacement against Hypergeometric sampling without replacement, and extract the single-trial Bernoulli variance. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 factor.
01
Configure Single-Trial Success Parameter
Set the target Bernoulli success probability to p = 0.40 on the digital parameter console and verify that theoretical single-trial variance equals p(1-p) = 0.40 · 0.60 = 0.240.
02
Execute Primary Binomial Simulation Batch
Run an ensemble of 10,000 independent binomial experiments with trial factor k = 1.0, logging empirical success frequencies across all trial runs.
03
Compare with Hypergeometric Urn Sampling
Activate the finite population urn mode with N_pop = 500 items (200 successes, 300 failures). Execute sampling without replacement and measure the variance reduction factor.
04
Verify Convergence of the Hypergeometric to Binomial
Increase population size N_pop while holding ratio p = M/N constant, observing that the finite population correction factor (N - n)/(N - 1) approaches 1.000.
05
Step Trial Count Scaling Factor Across k = 1 to 6
Increment the trial factor k across standardized steps k ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the linear expansion of cumulative variance.
06
Perform Ordinary Least-Squares Parameter Estimation
Fit a linear regression line y = mx to the coordinate pairs (x = k, y = Var). Confirm that empirical slope m = 0.2400 precisely recovers the single-trial Bernoulli variance.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Binomial & Hypergeometric Sampling mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.2400 · x
| # | Independent Trial Count Factor k (factor) | Scaled Binomial Variance Var(X)/n (Var/n) | Trial Scale Factor k | Cumulative Variance Metric | Remove |
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