Probability Distributions and their Stories

• 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)