Example of a Cauchy distribution

Figure 3.11.

This shows an example of a Cauchy distribution with various parameters. We’ll generate the distribution using:

dist = scipy.stats.cauchy(...)

Where … should be filled in with the desired distribution parameters Once we have defined the distribution parameters in this way, these distribution objects have many useful methods; for example:

• dist.pmf(x) computes the Probability Mass Function at values x in the case of discrete distributions

• dist.pdf(x) computes the Probability Density Function at values x in the case of continuous distributions

• dist.rvs(N) computes N random variables distributed according to the given distribution

Many further options exist; refer to the documentation of scipy.stats for more details.

Code output:

Python source code:

# Author: Jake VanderPlas
# License: BSD
#   The figure produced by this code is published in the textbook
#   "Statistics, Data Mining, and Machine Learning in Astronomy" (2013)
#   For more information, see http://astroML.github.com
#   To report a bug or issue, use the following forum:
#    https://groups.google.com/forum/#!forum/astroml-general
import numpy as np
from scipy.stats import cauchy
from matplotlib import pyplot as plt

#----------------------------------------------------------------------
# This function adjusts matplotlib settings for a uniform feel in the textbook.
# Note that with usetex=True, fonts are rendered with LaTeX.  This may
# result in an error if LaTeX is not installed on your system.  In that case,
# you can set usetex to False.
if "setup_text_plots" not in globals():
    from astroML.plotting import setup_text_plots
setup_text_plots(fontsize=8, usetex=True)

#------------------------------------------------------------
# Define the distribution parameters to be plotted
gamma_values = [0.5, 1.0, 2.0]
linestyles = ['-', '--', ':']
mu = 0
x = np.linspace(-10, 10, 1000)

#------------------------------------------------------------
# plot the distributions
fig, ax = plt.subplots(figsize=(5, 3.75))

for gamma, ls in zip(gamma_values, linestyles):
    dist = cauchy(mu, gamma)

    plt.plot(x, dist.pdf(x), ls=ls, color='black',
             label=r'$\mu=%i,\ \gamma=%.1f$' % (mu, gamma))

plt.xlim(-4.5, 4.5)
plt.ylim(0, 0.65)

plt.xlabel('$x$')
plt.ylabel(r'$p(x|\mu,\gamma)$')
plt.title('Cauchy Distribution')

plt.legend()
plt.show()

[download source: fig_cauchy_distribution.py]