Discrete distributions

Discrete Uniform distribution

  • Story. A set of discrete outcomes that can be indexed with sequential integers each has equal probability, like rolling a fair die.
  • Example. A monkey can choose from any of \(n\) bananas with equal probability.
  • Parameters. The distribution is parametrized by the high and low allowed values.
  • Support. The Discrete Uniform distribution is supported on the set of integers ranging from \(y_{low}\) to \(y_{high}\), inclusive.
  • Probability mass function.

    \(\begin{align}
    f(y;y_\mathrm{low}, y_\mathrm{high}) = \frac{1}{y_\mathrm{high} - y_\mathrm{low} + 1}
    \end{align}\)

  • Usage

    Package Syntax
    NumPy np.random.randint(low, high+1)
    SciPy scipy.stats.randint(low, high+1)
    Stan categorical(theta), theta array with all equal values

  • Related distributions.
    • The Discrete Uniform distribution is a special case of the Categorical distribution where all \(θ_y\) are equal.
  • Notes.
    • This distribution is not included in Stan. Instead, use a Categorical distribution with equal probailities.
    • In SciPy, this distribution is know as scipy.stats.randint. The high parameter is not inclusive; i.e., the set of allowed values includes the low parameter, but not the high. The same is true for numpy.random.randint(), which is used for sampling out of this distribution.
params = [dict(name='low', start=0, end=10, value=0, step=1), 
          dict(name='high', start=0, end=10, value=10, step=1)] 
app = distribution_plot_app(x_min=0, 
                            x_max=10, 
                            scipy_dist=st.randint, 
                            params=params, 
                            transform=lambda low, high: (low, high+1), 
                            x_axis_label='y', 
                            title='Discrete continuous') 
bokeh.io.show(app, notebook_url=notebook_url)