Weibull, Exponential and Rayleigh Distribution

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The Weibull distribution is a continuous probability distribution named after the Swedish mathematician Waloddi Weibull

Weibull, Exponential and Rayleigh Distributions

The probability density function for the two parameter Weibull distribution is

Weibull, Exponential and Rayleigh Distribution formulas

where k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution.

A three parameter Weibull distribution involves allocation parameter, gamma. The pdf is similar to the two parameter pdf with x being replaced by x-gamma.

The Weibull distribution is used extensively in Reliability Analysis

If the quantity X is a "time-to-failure",

A value of k < 1 indicates that the failure rate decreases over time.

A value of k =1 indicates that the failure rate is constant

A value of k > 1 indicates that the failure rate increases over time.

When k=1 the distribution is an Exponential Distribution and when k=2 the distribution is a Rayleigh Distribution

Applications are in Reliability, Survival Analysis, Engineering, Weather Forecasting, Hydrology and others

One application for the Weibull or Rayleigh distribution are used to represent a probabilistic based model to estimate the wind power in a given region.

The exponential distribution is often relevant for applications where the amount of time to some specific event important, such as the next earth quake, or plane crash.

The BIS.Net Process Performance APP uses machine powered algorithms to determine the parameters such that the Anderson Darling statistic is minimized.

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