Stochastic Arrival Generator
Generates independent exponentially distributed inter-arrival times using the inverse probability transform T_i = -ln(U)/λ.
Statistics & Probability / Poisson Process & Waiting Times
FREE TO EXPLORELABORATORY 13 · STATISTICS & PROBABILITY
Poisson Process, Rate Parameter Estimation, and Exponential Inter-Arrival Times
Model continuous-time stochastic point processes, quantify event arrival counting distributions, test inter-arrival time exponentiality, and verify Poisson dispersion index equality. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
SAMPLE VARIANCE VAR(X)—
SAMPLING TELEMETRY
Ready to sample with Rate λ = 1.0.
Adjust Rate λ, 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
Real-world physical processes often violate the strict independence assumption due to dead-time effects or refractory periods. The empirical regression of sample variance against rate intensity λ yields a linear gradient of m = 1.0000, confirming Poisson equidispersion with R² = 1.000000.
Statistical tools & sampling modules.
Time-Window Event Accumulator
Counts total arrivals N(t) across disjoint temporal intervals to construct empirical frequency histograms.
Inter-Arrival Exponential Density Analyzer
Fits continuous exponential density functions f(t) = λ e^(-λt) to empirical waiting times.
Fano Factor Evaluator
Computes the dispersion index D = Var(N)/E[N] to test for equidispersed Poisson behavior (D = 1.0).
Non-Homogeneous Poisson Modulator
Applies sinusoidal intensity profiles λ(t) = λ_0 (1 + a sin(ωt)) to evaluate time-varying arrival rates.
Linear Equidispersion Regression Engine
Performs linear regression of sample variance versus rate parameter to confirm the unit slope m = 1.000.
FOLLOW THE METHOD
A reason behind every step.
Model continuous-time stochastic point processes, quantify event arrival counting distributions, test inter-arrival time exponentiality, and verify Poisson dispersion index equality. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 events/unit.
01
Initialize Intensity Parameter and Time Horizon
Set the baseline Poisson arrival intensity to λ = 1.0 events per unit time on the control panel, establishing a continuous observation window of T = 100 time units.
02
Simulate Arrival Stream via Exponential Increments
Generate a continuous arrival stream by drawing successive inter-arrival waiting times from the exponential distribution T_k ~ Exponential(λ) until cumulative time exceeds T.
03
Bin Observations into Uniform Time Intervals
Partition the total timeline into 100 unit-length bins. Count the number of arrivals falling in each bin and log the empirical count distribution.
04
Evaluate Equidispersion and Fano Factor
Compute the sample mean x̄ and sample variance s² of the binned counts. Verify that the Fano factor D = s² / x̄ approaches the theoretical value of 1.000.
05
Step Arrival Intensity Across λ = 1.0 to 6.0
Increment the intensity parameter across standardized settings λ ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the simultaneous growth of both mean and variance.
06
Execute Equidispersion Linear Regression
Perform linear regression on the coordinate pairs (x = λ, y = Var(X)). Confirm that the empirical gradient m = 1.0000 rigorously verifies the foundational Poisson property Var(X) = E[X] = λ.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Poisson Process & Waiting Times mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 1.0000 · x
| # | Poisson Rate Intensity λ (events/unit) | Sample Variance Var(X) (Var(X)) | Rate Intensity λ | Sample Variance Var(X) | Remove |
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