Probability Distributions and their Stories

• Story. The intercept on the x-axis of a beam of light coming from the point $(μ,σ)$ is Cauchy distributed. This story is popular in physics (and is one of the first examples of Bayesian inference in Sivia's book), but is not particularly useful. You can think of it as a peaked distribution with enormously heavy tails.
• Parameters. The Cauchy distribution is peaked, and its peak is located at $μ$. The peak's width is dictated by parameter $σ$.
• Support. The Cauchy distribution is supported on the set of real numbers.
• Probability density function.
  
  $\begin{align}f(y;\mu, \sigma) = \frac{1}{\pi \sigma}\,\frac{1}{1 + (y-\mu)^2/\sigma^2}\end{align}$
  
  
• Usage
  
  

  Package	Syntax
  NumPy	mu + sigma * np.random.standard_cauchy()
  SciPy	scipy.stats.cauchy(mu, sigma)
  Stan	cauchy(mu, sigma)

  
  
• Related distributions.
• The Cauchy distribution is a special case of the Student-t distribution in which the degrees of freedom $ν = 1$.
• The numpy.random module only has the Standard Cauchy distribution ($μ=0$ and $σ=1$), but you can draw out of a Cauchy distribution using the transformation shown in the NumPy usage above.

params = [dict(name='µ', start=-0.5, end=0.5, value=0, step=0.01),
          dict(name='σ', start=0.1, end=1.0, value=0.2, step=0.01)]
app = distribution_plot_app(x_min=-2,
                            x_max=2,
                            scipy_dist=st.cauchy,
                            params=params,  
                            x_axis_label='y',
                            title='Cauchy')
bokeh.io.show(app, notebook_url=notebook_url)