Weibull Distribution Calculator
Calculate Weibull PDF, CDF, reliability, and hazard rate from shape and scale parameters.
Find mean, median, and mode for reliability and failure analysis.
Weibull Distribution
The Weibull distribution is a versatile probability distribution widely used in reliability engineering, survival analysis, and failure time modelling. By adjusting the shape parameter k, it can model increasing, constant, or decreasing failure rates.
Parameters
- k (shape / β): Controls the failure rate behavior
- λ (scale / η): Characteristic life — the time by which 63.2% of units will have failed
Key Formulas
| Function | Formula |
|---|---|
| f(x) = (k/λ)(x/λ)^(k−1) × exp(−(x/λ)^k) | |
| CDF | F(x) = 1 − exp(−(x/λ)^k) |
| Reliability | R(x) = exp(−(x/λ)^k) |
| Hazard rate | h(x) = (k/λ)(x/λ)^(k−1) |
| Mean | μ = λ × Γ(1 + 1/k) |
| Median | m = λ × (ln 2)^(1/k) |
| Mode | (k−1)^(1/k) × λ/k^(1/k) for k > 1, else 0 |
Shape Parameter Interpretation
| k | Failure Rate | Common Scenario |
|---|---|---|
| k < 1 | Decreasing | Infant mortality / early failures |
| k = 1 | Constant | Random / exponential distribution |
| k = 2 | Increasing (linear) | Wear-out failures |
| k = 3–4 | Bell-shaped | Near-normal, aging products |
The 63.2% Rule
At x = λ (the scale parameter), CDF always equals 1 − e⁻¹ ≈ 63.2%, regardless of k. This makes λ a natural “characteristic life” reference point.
Applications
Wind speed distributions, bearing lifetimes, battery failure rates, cancer survival curves, and extreme weather modelling all use Weibull distributions.