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What Is the Weibull Distribution?
The Weibull distribution is a continuous probability distribution widely used in reliability engineering, failure analysis, and survival analysis. Named after Swedish mathematician Waloddi Weibull, it is extremely versatile because different shape parameters produce distributions resembling the exponential, Rayleigh, or approximately normal distributions.
Its primary application is modeling time-to-failure data in reliability engineering. By fitting a Weibull distribution to failure data, engineers can predict product lifetimes, schedule preventive maintenance, and estimate warranty costs. The shape parameter reveals whether the failure rate is increasing, constant, or decreasing over time.
Formulas
Shape Parameter Effects
| k Value | Failure Rate | Distribution Shape |
|---|---|---|
| k < 1 | Decreasing | Infant mortality (early failures) |
| k = 1 | Constant | Exponential (random failures) |
| k = 2 | Linear increase | Rayleigh distribution |
| k ≈ 3.6 | Increasing | Approximately normal |
| k > 3.6 | Rapidly increasing | Wear-out failures |
Frequently Asked Questions
How do I interpret the shape parameter?
The shape parameter k (or beta) tells you about the failure pattern. k less than 1 means early-life failures decrease over time (burn-in). k equal to 1 means random failures at a constant rate (exponential). k greater than 1 means wear-out failures that increase over time. Most mechanical components have k between 1.5 and 4.
What is the bathtub curve?
The bathtub curve describes the typical failure rate pattern over a product's lifetime: high initial failure rate (infant mortality, k less than 1), followed by a low constant rate (useful life, k=1), then increasing rate (wear-out, k greater than 1). The Weibull distribution can model each phase with different shape parameters.