Discrete distributions

Poisson distribution

  • 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)