What is a Weibull distribution?
The Weibull distribution is a standard model for lifetimes, failure times, and waiting times (x≥0).
- k=1: exponential-like.
- k<1: more mass near 0 (often “decreasing hazard”).
- k>1: a peak appears (often “increasing hazard”).
- λ scales the overall time/length (bigger λ → larger values).
PDF: f(x)=(k/λ)(x/λ)^(k-1)exp(-(x/λ)^k). Mean: λ·Γ(1+1/k). Variance: λ²(Γ(1+2/k)-Γ(1+1/k)²). Median: λ·(ln 2)^(1/k).
You don’t need to enter personal information to use this tool.
Presets
Quickly set common shapes (you can tweak values after applying).
Generator
Set k/λ, sample size, bins, and RNG. Then generate samples and export results.
Sample stats
Samples (first 20)
How to use this tool
Use this page for lifetimes, waiting times, and reliability scenarios where the failure pattern matters.
Use in 3 steps
- Set the shape
kand scaleλusing a rough lifetime scenario you can explain. - Generate a sample and compare the histogram with the theoretical mean, variance, and quantiles.
- Change either
korλ, but not both at once, so you can separate shape effects from scale effects.
How to read the result
The scale λ stretches the horizontal axis, while the shape k changes whether mass clusters near zero or shifts away from it. This matters when you compare early-failure and wear-out behavior.
Boundary checks
- If
k<1, density is strongest near 0 and early failures become more common. - If
k≈1, the model behaves like an exponential waiting-time model. - Do not mix up a scale change with a change in failure pattern.
Frequently asked questions
What is the difference between k and λ?
k controls the shape or failure pattern, while λ controls the overall time scale.
Why does the curve hug 0 when k is small?
For k below 1, the density becomes stronger near zero, which represents more early events or early failures.
Why might another tool give a different mean?
Different parameter conventions, units, rounding, or sample sizes can change reported summary values.
Is this a bounded probability model?
No. Weibull is for non-negative durations or magnitudes, not for values restricted to the 0–1 interval.
What should I do first on this page?
Start with one scale and compare only two shape values, such as k below 1 and k above 1.