Sequential Bernoulli Sampler
Generates independent binary trials sequentially until a specified success stopping condition is achieved.
Statistics & Probability / Geometric & Negative Binomial
FREE TO EXPLORELABORATORY 14 · STATISTICS & PROBABILITY
Geometric Distribution, Memoryless Property, and Negative Binomial Stopping Times
Analyze discrete waiting times until success in Bernoulli sequences, test the discrete memoryless property, model Negative Binomial stopping times, and determine the linear scaling of expected waiting time with stopping quota r. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EXPECTED TRIAL COUNT E[X]—
SAMPLING TELEMETRY
Ready to sample with Quota r = 1.0.
Adjust Quota r, 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 memoryless property fails if the success probability p shifts or drifts across sequential trials. The empirical regression of expected trial count against success quota r yields a gradient of m = 2.0000, confirming theoretical waiting time scaling with R² = 1.000000.
Statistical tools & sampling modules.
Geometric Waiting Time Counter
Records the number of trials required to achieve the very first success (r = 1).
Memoryless Property Verifier
Evaluates conditional probabilities P(X > s + t | X > s) to confirm discrete memoryless behavior.
Negative Binomial Quota Accumulator
Logs the total number of trials required to observe exactly r successes across Monte Carlo batches.
Pascal-Polya Formulation Engine
Calculates theoretical probabilities under both total-trial and failure-count parameterizations.
Stopping Time Regression Unit
Performs linear regression of expected trial count against success quota r to extract the slope 1/p = 2.000.
FOLLOW THE METHOD
A reason behind every step.
Analyze discrete waiting times until success in Bernoulli sequences, test the discrete memoryless property, model Negative Binomial stopping times, and determine the linear scaling of expected waiting time with stopping quota r. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 successes.
01
Calibrate Single-Trial Probability
Set the Bernoulli success probability to p = 0.50 on the simulation dashboard, noting that the theoretical expected trials until first success is 1/p = 2.000.
02
Execute Geometric Trial Series (Quota r = 1)
Simulate 10,000 runs of the experiment with stopping quota r = 1. Record the distribution of total trials required to obtain the initial success.
03
Empirically Verify the Discrete Memoryless Property
Compute conditional frequencies P(X > s + t | X > s) for s = 2 and t = 1, verifying agreement with the unconditional survival probability P(X > t) = (1-p)^t = 0.50.
04
Extend to Negative Binomial Sequences
Configure the stopping engine to halt only when exactly r successes have accumulated, observing that the Negative Binomial variable is the sum of r independent Geometric variables.
05
Step Required Success Quota Across r = 1 to 6
Increment the target quota across standardized settings r ∈ {1, 2, 3, 4, 5, 6}, recording the empirical mean total trials E[X] for each setting.
06
Conduct Linear Regression on Stopping Times
Fit a linear regression model y = mx to the coordinate pairs (x = r, y = E[X]). Confirm that the empirical gradient m = 2.0000 matches theoretical expectation E[X] = r/p.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Geometric & Negative Binomial mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 2.0000 · x
| # | Success Quota Target r (successes) | Expected Trial Count E[X] (Trials E[X]) | Success Quota r | Expected Total Trials E[X] | Remove |
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