Probability Distributions and their Stories

• Story. Distribution of \(x=y^β\) if \(y\) is exponentially distributed. For \(β>1\), the longer we have waited, the more likely the event is to come, and vice versa for \(β<1\).
• Example. This is a model for aging. The longer an organism lives, the more likely it is to die.
• Parameters. There are two parameters, both strictly positive: the shape parameter \(β\), which dictates the shape of the curve, and the scale parameter \(τ\), which dictates the rate of arrivals of the event.
• Support. The Weibull distribution has support on the positive real numbers.
• Probability density function.
  
  
  \(\begin{align}
  f(y;\alpha, \sigma) = \frac{\alpha}{\sigma}\left(\frac{y}{\sigma}\right)^{\alpha - 1}\,
  \mathrm{e}^{-(y/\sigma)^\alpha}.
  \end{align}\)
  
  
  
• Usage
  
  

  Package	Syntax
  NumPy	np.random.weibull(alpha) * sigma
  SciPy	`scipy.stats.weibull_min(alpha, loc=0, scale=sigma)
  Stan	weibull(alpha, sigma)

  
  
• Related distributions.
  • The special case where \(α=0\) is the Exponential distribution with parameter \(β=1/σ\).
• Notes.
  • SciPy has a location parameter, which should be set to zero, with \(β\) being the scale parameter.
  • NumPy only provides a version of the Weibull distribution with \(σ=1\). Sampling out of the Weibull distribution may be accomplished by multiplying the resulting samples by \(σ\).

params = [dict(name='α', start=0.1, end=5, value=1, step=0.01),
          dict(name='σ', start=0.1, end=3, value=1.5, step=0.01)]
app = distribution_plot_app(x_min=0,
                            x_max=8,
                            scipy_dist=st.weibull_min,
                            params=params,
                            transform=lambda a, s: (a, 0, s),
                            x_axis_label='y',
                            title='Weibull')
bokeh.io.show(app, notebook_url=notebook_url)