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Weibull distribution
Definition
The Weibull distribution is a flexible continuous distribution for nonnegative quantities, widely used for time-to-event and reliability models. In its common two-parameter form, a positive shape parameter controls the distribution's form and a positive scale parameter controls its time or measurement scale. The hazard can decrease, remain constant or increase depending on shape. A basic two-parameter Weibull does not produce every possible hazard pattern, such as a full bathtub curve.
What it means
A lifetime distribution describes how event times vary across a population, not the exact lifetime of a particular item. Weibull's shape flexibility makes it useful for several kinds of event-time behavior. Shape below one gives a decreasing hazard, shape equal to one gives the exponential model with constant hazard, and shape above one gives an increasing hazard. These are mathematical properties of the model; observing a fitted shape does not by itself prove a specific physical failure mechanism.
How it works
With shape k and scale lambda, the two-parameter survival function is S(t) = exp[-(t/lambda)^k] for t at least zero. The CDF is one minus S(t), and the hazard is (k/lambda)(t/lambda)^(k minus 1) for positive t. The p quantile is lambda[-log(1 minus p)]^(1/k). A three-parameter version adds a location shift and has different estimation considerations. Fit the model using a method appropriate to complete or censored data, then assess whether its shape and predictions describe the observed process. Always check software parameter conventions.
Three practical examples
Example 1: A reliability team models time to component failure while including units still operating at study end. Example 2: A duration analysis uses Weibull event times as one candidate parametric model. Example 3: A simulation generates positive waiting times with an increasing modeled hazard. These examples are modeling choices, not evidence that every lifetime follows Weibull. Mixtures, changing operating conditions and competing failure mechanisms may require a different or richer model.
Think of it this way
Think of a flexible curve describing how a population of items gradually leaves service. One setting allows failures to be concentrated early, another produces a constant instantaneous rate, and another makes the rate rise with age. The curve summarizes a population pattern; it does not act as a countdown clock attached to every individual item.
Worked example
Suppose lifetime T follows a Weibull distribution with shape two and scale 100 hours. Survival to fifty hours is exp[-(50/100)^2] = exp(-0.25), approximately 0.7788. By 100 hours, the failure probability is one minus exp(-1), approximately 0.6321. The median lifetime is 100 sqrt(log 2), approximately 83.26 hours. The hazard is 2t/10,000, so it equals 0.01 per hour at fifty hours and 0.02 per hour at 100 hours. Among items that have survived fifty hours, the probability of surviving to 100 is S(100)/S(50) = exp(-0.75), approximately 0.4724. This conditional probability differs from unconditional survival to 100, illustrating that the shape-two model is not memoryless.
Illustrative case study
In this fictional case study, Acorn Equipment evaluates a Weibull model for component operating times. Analysts include right-censored units in the likelihood rather than treating them as failures or deleting them. They compare the fitted survival curve with data and examine whether different operating environments should be modeled separately. The report explains the shape and scale convention and gives uncertainty around relevant lifetime quantiles. Managers do not interpret the scale parameter as the average lifetime or a guaranteed service period.
Common misconceptions and mistakes
Common mistakes include treating scale as the mean, assuming every Weibull has an increasing hazard, and confusing hazard with a probability over a whole interval. Another mistake is discarding censored observations or assuming the basic model can reproduce any failure pattern. In the two-parameter model, the scale is the point at which cumulative failure probability is about 63.2%, regardless of shape. Fitted parameters also depend on the chosen units and parameterization, so formulas should not be transferred blindly between software packages.
Business use
Businesses can use Weibull models for planning and reliability analysis when the model and data collection are credible. Report the event definition, exposure units, censoring treatment and operating population. Check tail predictions carefully when decisions depend on rare early failures or long service life. Distinguish uncertainty in estimated parameters from variability among individual lifetimes. A fitted curve can support comparison and planning, but extrapolation far beyond observed data remains dependent on the model assumptions.
People also ask
Is the scale parameter generally the mean?
No.
What shape gives an exponential distribution?
One.
Can shape below one give decreasing hazard?
Yes.
Does hazard have probability units?
It is an instantaneous event rate, such as per hour.
Can the basic two-parameter Weibull represent a full bathtub hazard?
No; its hazard is monotone or constant.
Check your understanding
What fraction is expected to survive fifty hours in the example?
Answer: Approximately 0.7788.
What is the median lifetime?
Answer: Approximately 83.26 hours.
What failure probability corresponds to the scale of 100 hours?
Answer: Approximately 0.6321.
Why is survival from fifty to 100 hours calculated as a ratio?
Answer: It conditions on having already survived to fifty.
Related terms
- Hazard rate
- Censoring
- Related terms include survival function
- exponential distribution
- characteristic life
- quantile and reliability
Learn next
Learn next how censored observations contribute survival probabilities to a likelihood. Compare Weibull probability plots with model-based diagnostic checks, then examine alternative survival distributions and mixtures. Practice interpreting conditional survival and hazard separately before using a fitted curve for operational decisions.
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Sources
Source: NIST/SEMATECH Engineering Statistics Handbook, Weibull Distribution, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm. This primary reference defines the distribution, parameterization and related functions. The shape-two survival, hazard and quantile calculations are original and independently checked. The equipment case is fictional, and its parameters are illustrative rather than measured reliability claims.
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