2.2.5.10. Cauchy law

This law describes a Cauchy-Lorentz distribution with a location parameter \(x_0\) and a scale parameter \(\gamma\), as

\[f(x) = \frac{\gamma}{\pi\times(\gamma^{2}+(x-x_{0})^{2})}\]

The parameter \(\gamma\) should be greater than 0.0001.

Uranie code to simulate a Cauchy random variable is:

tds = DataServer.TDataServer("tdssampler", "Sampler Uranie demo")
tds.addAttribute(DataServer.TCauchyDistribution("cau", 0.3, 1.0))

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

tds.Draw("cau")

Figure 2.17 shows the PDF, CDF and inverse CDF generated for different sets of parameters.

../../../_images/TCauchyDistribution.png — Figure 2.17 Example of PDF, CDF and inverse CDF for Cauchy distributions.

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

tds.getAttribute("cau").setBounds(-1.0,2.0)  # truncate the law

The resulting PDF, CDF and inverse CDF, with and without truncation, can be seen, in this case, in Figure 2.18 for a given set of parameters and various boundaries.