- 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)