Weibull Random Variate Generator
Generates independent failure times using the inverse CDF transform T = η (-ln(1 - U))^(1/β).
Statistics & Probability / Weibull Reliability & Hazard
FREE TO EXPLORELABORATORY 19 · STATISTICS & PROBABILITY
Weibull Distribution, Hazard Rate Analysis, and Reliability Engineering
Model failure time distributions, evaluate the continuous Weibull distribution, analyze monotone hazard functions (aging, burn-in, constant failure rate), and extract the Weibull shape parameter β from log-log reliability data. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
CUMULATIVE HAZARD METRIC -LN(R(T))—
SAMPLING TELEMETRY
Ready to sample with Time t = 2.0.
Adjust Time t, 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
Simple linear regression on Weibull probability paper introduces slight finite-sample bias relative to maximum likelihood estimation. The log-log regression of ln(-ln(R(t))) against ln(t) yields a linear slope of m = 2.0000, confirming the theoretical shape parameter β = 2.000 with R² = 1.000000.
Statistical tools & sampling modules.
Kaplan-Meier Reliability Estimator
Computes empirical survival curves R(t) = P(T > t) from observed component failure times.
Instantaneous Hazard Rate Analyzer
Calculates the instantaneous hazard rate h(t) = f(t)/R(t) = (β/η)(t/η)^(β - 1) to classify component aging.
Bathtub Curve Simulator
Combines infant mortality (β < 1), useful life (β = 1), and wear-out (β > 1) regimes into a composite hazard profile.
Weibull Probability Paper Plotter
Linearizes survival data onto coordinates x = ln(t) and y = ln(-ln(R(t))) for graphical parameter estimation.
Log-Log Reliability Regression Unit
Performs linear regression on transformed coordinates to extract shape parameter β = 2.000 and scale η.
FOLLOW THE METHOD
A reason behind every step.
Model failure time distributions, evaluate the continuous Weibull distribution, analyze monotone hazard functions (aging, burn-in, constant failure rate), and extract the Weibull shape parameter β from log-log reliability data. Standardized parameter settings: 2.0, 3.0, 4.0, 5.0, 6.0, 7.0 hours.
01
Set Weibull Shape and Scale Parameters
Configure shape parameter β = 2.0 (Rayleigh wear-out regime) and scale parameter η = 2.5 on the reliability simulator dashboard.
02
Simulate Component Failure Times
Draw 50,000 independent failure times using the inverse probability formula T = 2.5 · (-ln(1 - U))^(1/2). Log the failure timestamps.
03
Calculate Empirical Survival Function R(t)
Sort failure times and compute the empirical reliability function R(t) = (Number of surviving units at time t) / (Total initial units).
04
Evaluate the Instantaneous Hazard Function
Compute the hazard function h(t) = (β/η)(t/η)^(β - 1). For β = 2.0, observe that h(t) = (2 / 6.25) t increases linearly with time, confirming positive aging.
05
Step Observation Time Across Standardized Grid t = 2 to 7
Evaluate survival probabilities across standardized inspection times t ∈ {2.0, 3.0, 4.0, 5.0, 6.0, 7.0}, computing transformed coordinates x = ln(t) and y = ln(-ln(R(t))).
06
Conduct Log-Log Linear Regression Analysis
Fit a linear regression model y = mx + b to the log-log reliability coordinates. Confirm that empirical gradient m = 2.0000 extracts the exact shape parameter β = 2.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Weibull Reliability & Hazard mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
ln(-ln(R(t))) = 2.0000 · ln(t) - 1.8326
| # | Operating Lifetime t (hours) | Cumulative Hazard Metric -ln(R(t)) (Hazard H(t)) | Log Time ln(t) | Log Cumulative Hazard ln(-ln(R(t))) | Remove |
|---|
No readings yet. Execute a statistical trial and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING