Continuous distributions

Gamma distribution

  • 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)