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In probability theory and statistics, the Weibull distribution / ˈ w aɪ b ʊ l / is a continuous probability distribution. 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 Fréchet distribution, also known as inverse Weibull distribution, [2] [3] is a special case of the generalized extreme value distribution. It has the cumulative distribution function ( ) = > . where α > 0 is a shape parameter.
Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables. In some fields of application the generalized extreme value distribution is known as the Fisher–Tippett distribution, named after R.A. Fisher and L.H.C. Tippett who recognised three different forms outlined ...
The commonly-used Weibull distribution combines both of these effects, as do the log-normal and hypertabastic distributions. After modelling a given distribution and parameters for h ( t ) {\displaystyle h(t)} , the failure probability density f ( t ) {\displaystyle f(t)} and cumulative failure distribution F ( t ) {\displaystyle F(t)} can be ...
The Weibull distribution or Rosin–Rammler distribution is a useful distribution for representing particle size distributions generated by grinding, milling and crushing operations. The log-hyperbolic distribution was proposed by Bagnold and Barndorff-Nielsen [9] to model the particle-size distribution of naturally occurring sediments. This ...
The Rice distribution is a noncentral generalization of the Rayleigh distribution: () = (,). The Weibull distribution with the shape parameter k = 2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter σ {\displaystyle \sigma } is related to the Weibull scale parameter according to λ = σ 2 . {\displaystyle \lambda =\sigma ...
The Weibull modulus is a dimensionless parameter of the Weibull distribution. It represents the width of a probability density function (PDF) in which a higher modulus is a characteristic of a narrower distribution of values.
The Gumbel distribution is a particular case of the generalized extreme value distribution (also known as the Fisher–Tippett distribution). It is also known as the log-Weibull distribution and the double exponential distribution (a term that is alternatively sometimes used to refer to the Laplace distribution).