Weibull distribution

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The Weibull distribution is a continuous probability distribution which is bounded below and unbounded above. It has three parameters\[a\], location; \(b>0\), scale; \(c>0\), shape. Its probability density function is given by

\(\frac{c}{b}\left(\frac{x-a}{b}\right)^{c-1}\exp\left(-\left(\frac{x-a}{b}\right)^{c}\right),\;\;\;x>a\).

The Weibull distribution is useful for modelling extreme values, representing the distribution of the maximum value out of a large number of random variables drawn from a bounded distribution.

Mean = \(a+b\Gamma\left(\frac{c+1}{c}\right)\) and Variance = \(b^2\left(\Gamma\left(\frac{c+2}{c}\right)-\Gamma\left(\frac{c+1}{c}\right)^2\right)\)

where \(\Gamma(\cdot)\) is the Gamma function.

References

  • Spiegel, MR, Schiller, JJ, Srinivasan, RA 2000 Schaum's Outline of Probability and Statistics. McGraw-Hill