Probability Distributions and their Stories

• Story. Rare events occur with a rate $λ$ per unit time. There is no "memory" of previous events; i.e., that rate is independent of time. A process that generates such events is called a Poisson process. The occurrence of a rare event in this context is referred to as an arrival. The number $n$ of arrivals in unit time is Poisson distributed.
• Example. The number of mutations in a strand of DNA per unit length (since mutations are rare) are Poisson distributed.
• Parameter. The single parameter is the rate $λ$ of the rare events occurring.
• Support. The Poisson distribution is supported on the set of nonnegative integers.
• Probability mass function.
  
  
  $\begin{align}f(n;\lambda) = \frac{\lambda^n}{n!}\,\mathrm{e}^{-\lambda}\end{align}$.
  
  
  
• Usage



Package	Syntax
NumPy	np.random.poisson(lam)
SciPy	scipy.stats.poisson(lam)
Stan	poisson(lam)




• Related distributions.
  
  • In the limit of $N→∞$ and $θ→0$ such that the quantity $Nθ$ is fixed, the Binomial distribution becomes a Poisson distribution with parameter $Nθ$. Thus, for large $N$ and small $θ$,
  $\begin{align}\\ \phantom{blah}f_\mathrm{Poisson}(n;\lambda) \approx f_\mathrm{Binomial}(n;N, \theta)\\ \phantom{blah}\end{align}$,
  with $λ=Nθ$. Considering the biological example of mutations, this is Binomially distributed: There are $N$ bases, each with a probability $θ$ of mutation, so the number of mutations, n is binomially distributed. Since $θ$ is small and $N$ is large, it is approximately Poisson distributed.
  
  
  • Under the $(μ,ϕ)$ parametrization of the Negative Binomial distribution, taking the limit of large $ϕ$ yields the Poisson distribution.

params = [dict(name='λ', start=1, end=20, value=5, step=0.1)] 
app = distribution_plot_app(x_min=0, 
                            x_max=40, 
                            scipy_dist=st.poisson,
                            params=params, 
                            x_axis_label='n', 
                            title='Poisson') 
bokeh.io.show(app, notebook_url=notebook_url)