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The Weibull distribution is a "waiting time" distribution where the hazard rate (i.e. the conditional probability of an event occurring, given that it has not already occurred), is a power of time with exponent α − 1. The parameter α is a shape parameter, and the parameter β is a scale parameter.

Parameter Range Description
α α > 0 Shape parameter
β β > 0 Scale parameter

Probability Density Function

f ( x ; α , β ) = α β ( x β ) α − 1 exp [ − ( x β ) α ]
F ( x ; α , β ) = 1 − exp [ − ( x β ) α ]
S ( x ; α , β ) = exp [ − ( x β ) α ]
h ( x ; α , β ) = α β ( x β ) α − 1

Support

0 ≤ x < ∞

Mean

Variance

Example α β
Failure rates for semiconductor chips often follow a Weibull distribution with α < 1, indicating that most failures occur early on. 0.6000 0.5000
The mean lifespan of a CFL bulb is 9000 hours. Let X be the time to failure of a randomly selected bulb (in 1000 hours). 3.520 10.00
Annual maximum discharge rates for the Feather River in California from 1902 - 1960 follow an approximate Weibull(1.30, 2.14) distribution. 1.300 2.140

X ∼ Weibull(α, β)

α = β =
Chart of the Weibull distribution Chart area for displaying the Weibull pdf, cdf, survival, hazard, visualization, and simulation

E(X) = , Var(X) =

The pdf can take several shapes depending on the parameter α.

  • When α < 1, the probability of an event occurring decreases with time and the pdf is strictly decreasing.
  • When α = 1, the probability of an event is independent of time and the pdf is exponentially decreasing.
  • When α > 1, the probability of an event occurring increases with time. The pdf is unimodal and can be either right-skewed, approximately symmetric, or left-skewed.

The pdf can take several shapes depending on the parameter α.

  • When α < 1, the probability of an event occurring decreases with time and the distribution is strictly decreasing.
  • When α = 1, the probability of an event is independent of time.
  • When α > 1, the probability of an event occurring increases with time. The distribution is unimodal and can be either right-skewed, approximately symmetric, or left-skewed.

The graph above displays the survival function S(x) = P(X > x) = 1 - F(X), where F(x) is the cumulative distribution function (cdf).

Survival functions are used in survival analysis, a branch of statistics concerned with the expected duration until an event occurs such as death or the failure of a mechanical system.

The graph above displays the hazard function h(x). This equals f(x)/S(x), where f(x) is the pdf and S(x) = P(X > x) is the survival function.

Note that:

  • When α < 1, the hazard function decreases with time.
  • When α = 1, the hazard function is constant and independent of time.
  • When α > 1, the hazard function increases with time.
  • The scale parameter β affects the scale but not the shape of the hazard function.

The graph above describes an event occuring randomly in time, where the conditional probability of an event given that it has not already occurred is given by the hazard function h(x) = (α/β)(x/β)α − 1. The time at which the first event occurs has a Weibull(α, β) distribution.

The simulation above shows an event occuring randomly in time, where the conditional probability of an event given that it has not already occurred is given by the hazard function h(x) = (α/β)(x/β)α − 1. The light blue line shows the time X at which the first event occurs, which has a Weibull(α, β) distribution. The histogram accumulates the results of each simulation.

Y = Weibull(1, β) ∼ Exponential(β) 1/X ∼ Fréchet(α, 1/β) −log(X) ∼ Gumbel(−log β, 1/α) min Xᵢ ∼ Weibull(α, βn−1/α)

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) = , Var(Y) =

The Weibull(α, β) distribution is shown in red superimposed over the exponential. It can be seen that these are identical when α = 1. This is the memoryless case. The probability of an event occurring is then independent of the time already spent waiting, which defines an exponential distribution.

This transformation motivates calling the Fréchet the inverse Weibull distribution, since a Fréchet random variable is the reciprocal of a Weibull random variable.

Note that this transformation maps the shape of the Weibull to the scale of the Gumbel and the scale of the Weibull to the location of the Gumbel.

The Weibull(α, β) distribution is shown in red superimposed over the distribution of the sample minimum. Note that:

  • Since α does not change, taking the minimum just changes the scale β of the distribution but not the shape.
  • Since a Weibull random variable is the reciprocal of a Fréchet random variable, the maximum of a sample of i.i.d Fréchet random variables is also Fréchet.