- 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)