Probability Distributions and their Stories

• Story. The story of the Student-t distribution largely derives from its relationships with other distributions. One way to think about it is as a Gaussian-like distribution with heavier tails.
• Parameters. The Student-t distribution is peaked, and its peak is located at $μ$. The peak's width is dictated by parameter σ. Finally, we define the "degrees of freedom" as $ν$. This last parameter imparts the distribution with heavy tails.
• Support. The Student-t distribution is supported on the set of real numbers.
• Probability density function.
  $\begin{align}f(y;\mu, \sigma, \nu) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\pi \nu \sigma^2}}\,\left(1 + \frac{(y-\mu)^2}{\nu\sigma^2}\right)^{-\frac{\nu+1}{2}}\end{align}$
  
  
• Usage



Package	Syntax
NumPy	mu + sigma * np.random.standard_t(nu)
SciPy	scipy.stats.t(nu, mu, sigma)
Stan	student_t(nu, mu, sigma)



• Related distributions.
  • In the $n→∞$ limit, the Student-t distribution becomes as Gaussian distribution.
  • The Cauchy distibution is a special case of the Student-t distribution in which $ν=1$.
  • Only the standard Student-t distribution ($μ=0$ and $σ=1$) is available in the numpy.random module. You can still draw out of the Student-t distribution by performing a transformation on the samples out of the standard Student-t distribution, as shown in the usage, above.
  • We get this distribution whenever we marginalize an unknown $σ$ out of a Gaussian distribution with an improper prior on $σ$ of $1/σ$.

params = [dict(name='ν', start=1, end=50, value=10, step=0.01),
          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.t,
                            params=params,  
                            x_axis_label='y',
                            title='Student-t')
bokeh.io.show(app, notebook_url=notebook_url)