Zero-Inflated Poisson Distribution#

Univariate, Discrete, Non-Negative, Zero-inflated

The Zero-Inflated Poisson (ZIP) distribution is a discrete probability distribution used to model count data characterized by an excess of zeros. It combines two components: a standard Poisson distribution and an additional mechanism that increases the probability of observing zero outcomes.

This distribution is particularly useful for scenarios where the data exhibit more zeros than what the Poisson model alone would predict. For example, in ecological studies, when researchers survey multiple habitats for a particular species, they often encounter many sites with zero observations (typically due to unsuitable habitat conditions) alongside a smaller number of sites where the species is observed in varying counts.

Key properties and parameters#

Support

\(x \in \mathbb{N}_0\)

Mean

\(\psi \mu\)

Variance

\(\psi \mu (1+(1-\psi) \mu\)

Parameters:

  • \(\psi\) : (float) Expected proportion of Poisson variates, \(0 \leq \psi \leq 1\).

  • \(\mu\) : (float) The mean rate of events of the Poisson component, \(\mu > 0\).

Probability Mass Function (PMF)#

\[\begin{split} f(x \mid \psi, \mu) = \left\{ \begin{array}{l} (1-\psi) + \psi e^{-\mu}, \text{if } x = 0 \\ \psi \frac{e^{-\mu}\mu^x}{x!}, \text{if } x=1,2,3,\ldots \end{array} \right. \end{split}\]
/home/docs/checkouts/readthedocs.org/user_builds/preliz/envs/stable/lib/python3.11/site-packages/pytensor/link/numba/dispatch/basic.py:211: UserWarning: Numba will use object mode to run XlogY0's perform method. Set `pytensor.config.compiler_verbose = True` to see more details.
  warnings.warn(

/home/docs/checkouts/readthedocs.org/user_builds/preliz/envs/stable/lib/python3.11/site-packages/pytensor/link/numba/dispatch/basic.py:211: UserWarning: Numba will use object mode to run XlogY0's perform method. Set `pytensor.config.compiler_verbose = True` to see more details.
  warnings.warn(

/home/docs/checkouts/readthedocs.org/user_builds/preliz/envs/stable/lib/python3.11/site-packages/pytensor/link/numba/dispatch/basic.py:211: UserWarning: Numba will use object mode to run XlogY0's perform method. Set `pytensor.config.compiler_verbose = True` to see more details.
  warnings.warn(
Zero-Inflated Poisson Distribution PMF

Cumulative Distribution Function (CDF)#

\[\begin{split} F(x \mid \psi, \mu) = \left\{ \begin{array}{l} (1-\psi) + \psi e^{-\mu}, \text{if } x = 0 \\ (1-\psi) + \psi \sum_{k=0}^x \frac{e^{-\mu}\mu^k}{k!}, \text{if } x=1,2,3,\ldots \end{array} \right. \end{split}\]
Zero-Inflated Poisson Distribution CDF

See also

Common Alternatives:

Related Distributions:

References#