Probability Distributions and their Stories

• Story. The amount of time we have to wait for $α$ arrivals of a Poisson process. More concretely, if we have events, $X_1$, $X_2$, …, $X_α$ that are exponentially distributed, $X_1+X_2+⋯+X_α$ is Gamma distributed.
• Example. Any multistep process where each step happens at the same rate. This is common in molecular rearrangements.
• Parameters. The number of arrivals, $α$, and the rate of arrivals, $β$.
• Support. The Gamma distribution is supported on the set of positive real numbers.
• Probability density function.
  
  
  $\begin{align}f(y;\alpha, \beta) = \frac{1}{\Gamma(\alpha)}\,\frac{(\beta y)^\alpha}{y}\,\mathrm{e}^{-\beta y}\end{align}$
  
  
  
• Related distributions.
  • The Gamma distribution is the continuous analog of the Negative Binomial distribution.
  • The special case where $α=1$ is an Exponential distribution.
  • The special case where $α=ν/2$ and $β=1/2$ is a Chi-square distribution parametrized by $ν$.
• Usage
  
  

  Package	Syntax
  NumPy	np.random.gamma(alpha, beta)
  SciPy	scipy.stats.gamma(alpha, loc=0, scale=beta)
  Stan	gamma(alpha, beta)

  
  
• Notes.
  • The Gamma distribution is useful as a prior for positive parameters. It imparts a heavier tail than the Half-Normal distribution (but not too heavy; it keeps parameters from growing too large), and allows the parameter value to come close to zero.
  • SciPy has a location parameter, which should be set to zero, with $β$ being the scale parameter.

params = [dict(name='α', start=1, end=5, value=2, step=0.01),
          dict(name='β', start=0.1, end=5, value=2, step=0.01)]
app = distribution_plot_app(x_min=0,
                            x_max=50,
                            scipy_dist=st.gamma,
                            params=params,
                            transform=lambda a, b: (a, 0, b),
                            x_axis_label='y',
                            title='Gamma')
bokeh.io.show(app, notebook_url=notebook_url)