Weibull Distribution

The Weibull distribution is given by

			(1)
			(2)

for , and is implemented in the Wolfram Language as WeibullDistribution[alpha, beta]. The raw moments of the distribution are

			(3)
			(4)
			(5)
			(6)

and the mean, variance, skewness, and kurtosis excess of are

			(7)
			(8)
			(9)
			(10)

where  is the gamma function and

(11)

A slightly different form of the distribution is defined by

			(12)
			(13)

(Mendenhall and Sincich 1995). This has raw moments

			(14)
			(15)
			(16)
			(17)

so the mean and variance for this form are

			(18)
			(19)

The Weibull distribution gives the distribution of lifetimes of objects. It was originally proposed to quantify fatigue data, but it is also used in analysis of systems involving a "weakest link."

REFERENCES:

Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 2, 2nd ed. New York: Wiley, 1995.

Kobayashi, A. (Ed.). Handbook on Experimental Mechanics. New York: VCH/SEM, 1993.

Mendenhall, W. and Sincich, T. Statistics for Engineering and the Sciences, 4th ed. Englewood Cliffs, NJ: Prentice Hall, 1995.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 119, 1992.