Weibull Distribution Calculator
Enter shape (k) and scale (λ) parameters to compute Weibull PDF, CDF, survival probabilities, and descriptive statistics.
PDF f(x): 0.167442
CDF F(x): 0.302324
Survival S(x): 0.697676
Mean: 4.4311
Variance: 5.3650
Mode: 3.5355
Density samples
| x | f(x) |
|---|---|
| 0.00 | 0.000000 |
| 2.08 | 0.140104 |
| 4.17 | 0.166451 |
| 6.25 | 0.104806 |
| 8.33 | 0.041451 |
| 10.42 | 0.010861 |
| 12.50 | 0.001930 |
| 14.58 | 0.000236 |
| 16.67 | 0.000020 |
| 18.75 | 0.000001 |
| 20.83 | 0.000000 |
| 22.92 | 0.000000 |
| 25.00 | 0.000000 |
How to Use This Calculator
- Specify the shape (k) and scale (λ) parameters from your reliability or life data model.
- Enter an x-value to compute the density, cumulative probability, and survival function.
- Use the summary statistics to describe expected life and dispersion.
- Leverage the density samples for visualization or further analysis.
Formula
f(x; k, λ) = (k / λ) (x / λ)k − 1 e−(x/λ)k, x ≥ 0
F(x) = 1 − e−(x/λ)k
Mean = λ Γ(1 + 1/k)
Variance = λ² [Γ(1 + 2/k) − Γ(1 + 1/k)²]
Γ(·) denotes the gamma function. The Weibull distribution generalizes exponential (k = 1) and Rayleigh (k = 2) cases.
Full Description
Weibull distributions are a versatile family for modeling time-to-failure data, with the shape parameter controlling failure rate behavior (increasing, constant, or decreasing hazard). Engineers rely on Weibull analyses to estimate product reliability, maintenance schedules, and warranty policies.
Shape k < 1 indicates high initial failure rates that decline; k = 1 corresponds to exponential lifetimes; k > 1 models wear-out mechanisms with increasing failure rates over time.
Frequently Asked Questions
How do I estimate Weibull parameters?
Use maximum likelihood estimation or linearized probability plots based on logged data.
What if x = 0?
The CDF evaluates to 0 and the PDF to (k / λ) · 0k − 1 (finite when k > 1, infinite otherwise). The calculator handles the limit numerically.
Is Weibull used outside reliability?
Yes. It appears in hydrology, meteorology, material strength modeling, and queueing theory.