Weibull Distribution
Print PageA Weibull Distribution Calculator (also known as a Weibull Reliability & Survival Calculator, Shape [k] & Scale [λ] Utility, Hazard Rate [h(t)] Analyzer, or Material Fatigue Life Calculator) computes exact Probability Density Function values (f(t) = [k ÷ λ] · (t ÷ λ)k-1 · exp( -[t ÷ λ]k ) for t ≥ 0), Cumulative Distribution Function failure probabilities (F(t) = 1 - exp( -[t ÷ λ]k )), Reliability Survival Function (R(t) = exp( -[t ÷ λ]k )), Instantaneous Hazard Rate (h(t) = [k ÷ λ] · (t ÷ λ)k-1), Mean Time Between Failures (MTBF = μ = λ · Γ(1 + 1/k)), median lifespan (t0.5 = λ · [&ln; 2]1/k), modal peak failure time (tmode), and variance.
In aerospace component testing, wind turbine bearing maintenance, electronic component burn-in, and biomedical implant reliability, the Weibull distribution models the bathtub curve of failure rates across infant mortality, random constant failures, and wear-out stages.
Our free online Weibull Distribution Calculator provides instant calculations across all continuous reliability parameters:
- Probability Density Function (PDF for t ≥ 0):
f(t; k, λ) = ( k ÷ λ ) · ( t ÷ λ )k-1 · e-( t ÷ λ )k. - Cumulative Distribution Function (CDF / Failure Probability F[t]):
F(t; k, λ) = 1 - e-( t ÷ λ )k. - Reliability / Survival Function (R[t] = P[T > t]):
R(t; k, λ) = e-( t ÷ λ )k. - Instantaneous Hazard Rate / Failure Rate (h[t]):
h(t) = ( k ÷ λ ) · ( t ÷ λ )k-1. - Bathtub Curve Shape Behavior (k):
k < 1: Infant mortality / early failure (Decreasing hazard rate).k = 1: Random exponential failure (Constant hazard rate1 ÷ λ).k > 1: Wear-out failure stage (Increasing hazard rate).k = 2: Rayleigh special distribution case.
- Characteristic Life Rule (t = λ): At
t = λ, exactly63.21%of units have failed (F(λ) = 1 - e-1 ≈ 0.63212). - Mean Time Between Failures (MTBF / μ):
μ = λ · Γ( 1 + 1 ÷ k ). - Median Lifespan (t0.5):
t0.5 = λ · [ &ln;(2) ]1/k.
Master Weibull Reference Table (Wind Turbine Bearing: Shape k = 2.50, Scale λ = 10,000 Hours)
The table below displays operating hours (t), reliability survival rates R(t), cumulative failure probabilities F(t), instantaneous hazard rates h(t), and maintenance statuses for a wind turbine bearing undergoing wear-out (Shape k = 2.50, Scale Characteristic Life λ = 10,000 Hours):
| Operating Hours (t Hours) | Hazard Rate h(t) per Hour | Reliability Survival R(t) | Cumulative Failure F(t) | Component Maintenance Status |
|---|---|---|---|---|
| 5,000 hours (Early Inspection) | 0.000088 per hour | 83.7960% (83.80%) | 16.2040% | 83.8% High Reliability Operating Zone |
| 8,152 hours (Mode / Peak Density) | 0.000185 per hour | 55.3340% (55.33%) | 44.6660% | Peak Failure Density Hour (t_mode) |
| 8,638 hours (Median Lifespan t0.5) | 0.000201 per hour | 50.0000% (50.00%) | 50.0000% (50.00%) | 50% Probability Survives Past 8,638 hrs |
| 8,873 hours (MTBF Mean μ) | 0.000209 per hour | 47.3820% (47.38%) | 52.6180% | Expected Average Bearing Lifespan |
| 10,000 hours (Characteristic Life λ) | 0.000250 per hour | 36.7879% (36.79%) | 63.2121% (63.21%) | Standard 63.21% Characteristic Life Rule |
| 12,000 hours (Overdue Replacement) | 0.000329 per hour | 20.6500% (20.65%) | 79.3500% (79.35%) | 79.4% High Wear-Out Failure Hazard |
Step-by-Step Wind Turbine Bearing Reliability Calculation
To evaluate component reliability for a wind turbine bearing experiencing wear-out with shape k = 2.50 and scale λ = 10,000 hours:
Step 1 (Identify Failure Stage from Shape k): k = 2.50 > 1.0 &implies; Wear-Out Stage (Increasing Hazard Rate h'[t] > 0)
Step 2 (Calculate Characteristic Life Failure at t = λ): F(10,000) = 1 - e^-(10,000 / 10,000)^2.5 = 1 - e^-1 = 1 - 0.36788 = 0.63212 ≈ 63.21%
Step 3 (Calculate MTBF Expected Mean Lifespan μ): μ = 10,000 · Γ(1 + 1/2.5) = 10,000 · Γ(1.40) = 10,000 · 0.88726 = 8,872.6 hrs ≈ 8,873 hrs
Step 4 (Calculate Median Lifespan t_0.5): t0.5 = 10,000 · (&ln;[2])^(1/2.5) = 10,000 · (0.69315)^0.40 = 10,000 · 0.86377 = 8,637.7 hrs ≈ 8,638 hrs
Step 5 (Calculate Survival Reliability at 5,000 Hours R[5,000]): R(5,000) = e^-(5,000 / 10,000)^2.5 = e^-(0.5)^2.5 = e^-0.17678 = 0.83796 ≈ 83.80%
Step 6 (Calculate Failure Probability by 12,000 Hours F[12,000]): F(12,000) = 1 - e^-(12,000 / 10,000)^2.5 = 1 - e^-(1.2)^2.5 = 1 - e^-1.57744 = 1 - 0.20650 = 0.79350 ≈ 79.35%
Thus, the bearing has an 83.80% probability of surviving 5,000 hours, an MTBF of 8,873 hours, and a 79.35% chance of failing before 12,000 hours.
Reliability Distributions Comparison: Weibull vs. Exponential vs. Rayleigh vs. Normal
Below is a comparative reference chart detailing when to use the Weibull distribution versus related reliability models:
| Reliability Model | Hazard Rate h(t) Profile | Shape Parameter Constraint | Primary Practical Application |
|---|---|---|---|
| Weibull Distribution | Flexible (Decreasing, Constant, or Increasing) | Flexible Shape k > 0 | Material fatigue, bearings, full bathtub curve modeling. |
| Exponential Distribution | Strictly CONSTANT Hazard Rate 1/λ (Memoryless) | Fixed Shape k = 1 | Electronic component random failures during useful life. |
| Rayleigh Distribution | Linear INCREASING Hazard Rate h(t) = 2t/λ2 | Fixed Shape k = 2 | Wind speed forecasting & RF signal fading. |
| Normal (Gaussian) Distribution | Symmetric Bell Curve (Non-monotone hazard) | Approximated by Weibull k ≈ 3.6 | Simple wear-out without early infant mortality. |
History & Mathematics: 1933 Rosin-Rammler to 1951 Waloddi Weibull
1933 Paul Rosin & Erich Rammler
In 1933, German engineers Paul Rosin and Erich Rammler first derived the mathematical function while modeling the particle size distribution of crushed coal in Journal of the Institute of Fuel.
1951 Waloddi Weibull & Engineering Applications
In 1951, Swedish engineer and mathematician Waloddi Weibull published his historic paper A Statistical Distribution Function of Wide Applicability in the ASME Journal of Applied Mechanics, demonstrating its immense power in predicting material fatigue strength, ball bearing life, and electrical insulation breakdown.
Popular direct tools:
Frequently Asked Questions (FAQ)
What is the Weibull Reliability Function formula?
The Reliability (Survival) Function is R(t) = exp( -[t ÷ λ]k ) for t ≥ 0.
What does the characteristic life (λ) mean in a Weibull distribution?
The scale parameter λ (characteristic life) is the operating time at which 63.21% of all units are expected to fail (F(λ) = 1 - e-1 ≈ 0.63212), regardless of the shape parameter k.
How does the shape parameter (k) indicate the failure mode?
If k < 1, the component exhibits infant mortality (early manufacturing defects). If k = 1, failures are random exponential events. If k > 1, the component exhibits age-related wear-out.