Weibull Distribution Formula
The Weibull distribution models time-to-failure and reliability.
Its flexible shape parameter makes it ideal for reliability engineering, wind speed analysis, and survival analysis.
The Formula
CDF: F(x; λ, k) = 1 − e^(−(x/λ)^k)
Mean: μ = λ Γ(1 + 1/k)
The Weibull distribution was introduced by Swedish engineer Waloddi Weibull in 1951. Its great strength is flexibility — by adjusting the shape parameter k, it can model decreasing, constant, or increasing failure rates. This makes it the most widely used distribution in reliability engineering and survival analysis.
Variables
| Symbol | Meaning | Note |
|---|---|---|
| x | Time or measurement value (x ≥ 0) | Often represents failure time |
| k | Shape parameter | k > 0 |
| λ | Scale parameter (characteristic life) | λ > 0 |
| Γ | Gamma function | Γ(n) = (n−1)! for integers |
Shape parameter k interpretation:
- k < 1: decreasing failure rate — "infant mortality" (early failures)
- k = 1: constant failure rate — reduces to exponential distribution (random failures)
- k = 2: linearly increasing rate — Rayleigh distribution (wear-out failures)
- k ≈ 3.5: approximates normal distribution (bell-shaped)
- k > 1: increasing failure rate — wear-out phase
Example 1 — Light Bulb Reliability
LED bulbs have Weibull parameters k = 2.5 and λ = 20,000 hours. What fraction survive to 10,000 hours?
Survival function S(t) = 1 − F(t) = e^(−(t/λ)^k)
S(10000) = e^(−(10000/20000)^2.5) = e^(−(0.5)^2.5)
0.5^2.5 = 0.5^2 × 0.5^0.5 = 0.25 × 0.7071 = 0.1768
S(10000) = e^(−0.1768) ≈ 0.838 — about 83.8% of LED bulbs survive to 10,000 hours
Example 2 — Rayleigh Distribution (k = 2)
Wind speed at a site follows Weibull with k = 2 (Rayleigh), λ = 8 m/s. Find the mean wind speed.
Mean = λ × Γ(1 + 1/k) = 8 × Γ(1.5)
Γ(1.5) = (1/2)! = √π/2 ≈ 0.8862
Mean wind speed = 8 × 0.8862 ≈ 7.09 m/s
When to Use It
Use the Weibull distribution when:
- Modeling time-to-failure of mechanical and electronic components
- Wind energy — wind speed at a given site follows Weibull distribution closely
- Medical survival analysis — time until patient relapse or death
- Material strength analysis — tensile strength of brittle materials
- Insurance and actuarial science — modeling time until an event