2.2.5.12. Weibull law

This law describes a weibull distribution depending on the location \(x_{\rm min}\), the scale \(\lambda\) and the shape \(k\), as

\[f(x) = \frac{k}{\lambda} \times\left(\frac{x-x_{\rm min}}{\lambda}\right)^{k-1} \times e^{-\left(\frac{x-x_{\rm min}}{\lambda}\right)^{k}} \; {\rm1\kern-0.28emI}_{[x_{\rm min},+\infty[}(x)\]

Both \(\lambda\) and \(k\) should be greater than 0.0001.

Uranie code to simulate a Weibull random variable is:

tds = DataServer.TDataServer("tdssampler", "Sampler Uranie demo")
tds.addAttribute(DataServer.TWeibullDistribution("wei", 0.5, 2.0, -0.01) )

fsamp = Sampler.TSampling(tds, "lhs", 300)
fsamp.generateSample()  # Create a representative sample

tds.Draw("wei")

Figure 2.21 shows the PDF, CDF and inverse CDF generated for different sets of parameters.

Figure 2.21 Example of PDF, CDF and inverse CDF for Weibull distributions.

Is it also possible to set boundaries to the infinite span of this distribution to create a truncated Weibull law. This can be done by calling the following method:

tds.getAttribute("wei").setBounds(0.2,1.8)  # truncate the law

The resulting PDF, CDF and inverse CDF, with and without truncation, can be seen, in this case, in Figure 2.22 for a given set of parameters and various boundaries.

Figure 2.22 Example of PDF, CDF and inverse CDF for a Weibull truncated distribution.