Probability Distributions and their Stories

• Story. We perform a series of Bernoulli trials with probability of success $θ$ until we get a success. The number of failures $y$ before the success is Geometrically distributed.
• Example. Consider actin polymerization. At each time step, an actin monomer may add to the filament ("failure"), or an actin monomer may fall off ("success") with (usually very low) probability $θ$. The length of actin filaments are Geometrically distributed.
• Parameter. The Geometric distribution is parametrized by a single value, $θ$, the probability that the Bernoulli trial is successful.
• Support. The Geometric distribution, as defined here, is has support on the nonnegative integers.
• Probability mass function.

• Usage

• Related distributions.
  • The Geometric distribution is a special case of the Negative Binomial distribution in which $α=1$ and $θ=β/(1+β)$.
  • The continuous analog of the Geometric distribution is the Exponential distribution.
• Notes.
  • The Geometric distribution is supported on non-negative integer $y$.
• The Geometric distribution is not implemented in Stan. You can instead use a Negative Binomial distribution fixing the parameter $α$ to be unity and relating the parameter $β$ of the Negative Binomial distribution to $θ$ as $θ=β/(1+β)$.
• The Geometric distribution is defined differently in SciPy, instead replacing $y$ with $y−1$. In SciPy's parametrization the Geometric distribution describes the number of successive Bernoulli trials (not just the failures; the success is included) necessary to get a success. To adjust for this, we use the loc=-1 kwarg.

params = [dict(name='θ', start=0, end=1, value=0.5, step=0.01)]
app = distribution_plot_app(x_min=0,
                            x_max=20,
                            scipy_dist=st.geom,
                            params=params,
                            transform=lambda theta: (theta, -1),
                            x_axis_label='y',
                            title='Geometric')
bokeh.io.show(app, notebook_url=notebook_url)