Generating Weibull Distributed Random Numbers

Generating Weibull Distributed Random Numbers

This is a step-by-step explaination of how to calculate a transformation function that converts a random variable of one distribution to another distribution. This example uses the Weibull distribution as the intended target distribution.

The Weibull distribution is defined as follows,

equation 1

The parameters alpha and beta are greater than zero. For values of x less than zero, the distribution is defined to be zero.

The mean for this distribution is:

equation 2

The variance is:

equation 3

To determine the transformation we start with the fundamental transformation law of probabilities for two probability distribution functions f(y) and p(x),

equation 4
equation 5

This allows us to relate a random variable x from the distribution function p(x) with the random variable y from the distribution function f(y). If x is from a uniform distribution in [0,1], then p(x) is constant so we have,

equation 6

The solution to this is the definite integral of f(y),

equation 7

This relationship gives us the source random variable, given the target random variable. So we need to be able to invert the relationship,

equation 8

It is the difficulty of the determination of y = G(x) given x = F(y) that makes this problem potentially infeasible.

For the case of the Weibull distribution we can readily determine the inverse.

Starting with,

equation 9

then doing this integral we get,

equation 10

Now we need to invert this to write, y = G(x),

equation 11

Finally, the transformation is:

equation 12

We start with the random number, x, which comes from a uniform distribution (in the range from 0 to 1). Then apply the above transformation to get a new independent random number which has a Weibull distribution with a mean and variance that depends upon the values of alpha and beta.

A Matlab Implementation

Here is a Matlab implementation of the above, contributed by David Vannuci from the University of the Witwatersrand,
function [WeibullRandomNumbers]=WeibullRNG(scale,shape,noOfRandomNumbers)
%Written by David Vannucci 01 June 2003
WeibullRandomNumbers = scale.*( -log(1-rand(noOfRandomNumbers,1))).^(1/shape);

Useful References

Knuth, D.E., 1981; The Art of Computer Programming, Volume 2 Seminumerical Algorithms, Addison-Wesley, Reading Mass., 688 pages, ISBN 0-201-03822-6

Press, W.H., B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, 1986; Numerical Recipes, The Art of Scientific Computing, Cambridge University Press, Cambridge, 818 pages, ISBN 0-512-30811-9

Everett (Skip) Carter
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