The Weibull distribution.

Create Weibull distribution from parameters.

If the shape (first) parameter is 1.0, the distribution is equivalent to a ExponentialDistribution with parameter 1 / lambda the scale (second) parameter.

Create Weibull distribution from parameters.

If the shape (first) parameter is 1.0, the distribution is equivalent to a ExponentialDistribution with parameter 1 / lambda the scale (second) parameter.

Standard Weibull distribution with scale factor (lambda) 1.

Create Weibull distribution from mean and standard deviation.

The algorithm is from "Methods for Estimating Wind Speed Frequency Distributions", C. G. Justus, W. R. Hargreaves, A. Mikhail, D. Graber, 1977. Given the identity:

\[ (\frac{\sigma}{\mu})^2 = \frac{\Gamma(1+2/k)}{\Gamma(1+1/k)^2} - 1 \]

\(k\) can be approximated by

\[ k \approx (\frac{\sigma}{\mu})^{-1.086} \]

\(\lambda\) is then calculated straightforwardly via the identity

\[ \lambda = \frac{\mu}{\Gamma(1+1/k)} \]

Numerically speaking, the approximation for \(k\) is accurate only within a certain range. We arbitrarily pick the range \(0.033 \le \frac{\sigma}{\mu} \le 1.45\) where it is good to ~6%, and will refuse to create a distribution outside of this range. The paper does not cover these details but it is straightforward to check them numerically.