Probability Distributions and their Stories

• Story. Any outcome in a given range has equal probability.
• Example. Anything in which all possibilities are equally likely. This is, perhaps surprisingly, rarely encountered.
• Parameters. The Uniform distribution is not defined on an infinite or semi-infinite domain, so lower and upper bounds, \(α\) and \(β\), respectively, are necessary parameters.
• Support. The Uniform distribution is supported on the interval \([α,β]\).
• Probability density function.
  
  
  \(\begin{align}
  f(y;\alpha, \beta) = \left\{ \begin{array}{ccc}
  \frac{1}{\beta - \alpha} & & \alpha \le y \le \beta \\(0.5em]
  0 & & \text{otherwise}
  \end{array}
  \right.
  \end{align}\)
  
  
  
• Usage
  
  

Package	Syntax
NumPy	np.random.uniform(alpha, beta)
SciPy	scipy.stats.uniform(alpha, beta)
Stan	uniform(alpha, beta)




• Related distributions.
• The Uniform distribution on the interval [0, 1] (i.e., \(α=0\) and \(β=1\)) is a special case of the Beta distribution where the parameters for the Beta distribution are \(α=β=1\) (not to be confused with the \(α\) and \(β\) used to parametrize the Uniform distribution).

params = [dict(name='α', start=-2, end=3, value=0, step=0.01),
          dict(name='β', start=-2, end=3, value=1, step=0.01)]
app = distribution_plot_app(x_min=-2,
                            x_max=3,
                            scipy_dist=st.uniform,
                            params=params,
                            title='Uniform')
bokeh.io.show(app, notebook_url=notebook_url)