It models a broad range of random variables, largely in the nature of a time to failure or time between events.

Examples are maximum one-day rainfalls and the time a user spends on a web page.

The distribution is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1939,[1][2] although it was first identified by René Maurice Fréchet and first applied by Rosin & Rammler (1933) to describe a particle size distribution.

The probability density function of a Weibull random variable is[3][4] where k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution.

[5] If the quantity, x, is a "time-to-failure", the Weibull distribution gives a distribution for which the failure rate is proportional to a power of time.

In the context of diffusion of innovations, the Weibull distribution is a "pure" imitation/rejection model.

Applications in medical statistics and econometrics often adopt a different parameterization.

Then, for x ≥ 0, the probability density function is the cumulative distribution function is the quantile function is and the hazard function is In all three parameterizations, the hazard is decreasing for k < 1, increasing for k > 1 and constant for k = 1, in which case the Weibull distribution reduces to an exponential distribution.

The form of the density function of the Weibull distribution changes drastically with the value of k. For 0 < k < 1, the density function tends to ∞ as x approaches zero from above and is strictly decreasing.

For k = 1, the density function tends to 1/λ as x approaches zero from above and is strictly decreasing.

For k > 1, the density function tends to zero as x approaches zero from above, increases until its mode and decreases after it.

Moreover, the skewness and coefficient of variation depend only on the shape parameter.

The failure rate h (or hazard function) is given by The Mean time between failures MTBF is The moment generating function of the logarithm of a Weibull distributed random variable is given by[12] where Γ is the gamma function.

Similarly, the characteristic function of log X is given by In particular, the nth raw moment of X is given by The mean and variance of a Weibull random variable can be expressed as and The skewness is given by where

The kurtosis excess may also be written as: A variety of expressions are available for the moment generating function of X itself.

As a power series, since the raw moments are already known, one has Alternatively, one can attempt to deal directly with the integral If the parameter k is assumed to be a rational number, expressed as k = p/q where p and q are integers, then this integral can be evaluated analytically.

The characteristic function has also been obtained by Muraleedharan et al. (2007).

The characteristic function and moment generating function of 3-parameter Weibull distribution have also been derived by Muraleedharan & Soares (2014) harvtxt error: no target: CITEREFMuraleedharanSoares2014 (help) by a direct approach.

be independent and identically distributed Weibull random variables with scale parameter

will also be Weibull distributed with scale parameter

The Weibull distribution is the maximum entropy distribution for a non-negative real random variate with a fixed expected value of xk equal to λk and a fixed expected value of ln(xk) equal to ln(λk) −

of data on special axes in a type of Q–Q plot.

The reason for this change of variables is the cumulative distribution function can be linearized: which can be seen to be in the standard form of a straight line.

There are various approaches to obtaining the empirical distribution function from data.

One method is to obtain the vertical coordinate for each point using where

[18][19] Another common estimator[20] is Linear regression can also be used to numerically assess goodness of fit and estimate the parameters of the Weibull distribution.

The gradient informs one directly about the shape parameter

The coefficient of variation of Weibull distribution depends only on the shape parameter:[21] Equating the sample quantities

can be found using a root finding algorithm to solve The moment estimate of the scale parameter can then be found using the first moment equation as The maximum likelihood estimator for the

is[citation needed] Again, this being an implicit function, one must generally solve for