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Concept# Weibull distribution

Summary

In probability theory and statistics, the Weibull distribution ˈwaɪbʊl 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 distribution is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1939, although it was first identified by Maurice René Fréchet and first applied by to describe a particle size distribution.
Definition
Standard parameterization
The probability density function of a Weibull random variable is
:
f(x;\lambda,k) =
\begin{cases}
\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}}, & x\geq0 ,\
0, & x 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function. Th

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