Poisson Distribution#

Univariate, Discrete, Non-Negative

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time (or space) if these events occur with a known constant mean rate and independently of the time since the last event.

Key properties and parameters#

Support

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

Mean

\(\mu\)

Variance

\(\mu\)

Parameters:

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

Probability Density Function (PDF)#

\[ f(x \mid \mu) = \frac{e^{-\mu}\mu^x}{x!} \]
/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(
Poisson Distribution PDF

Cumulative Distribution Function (CDF)#

\[ F(x \mid \mu) = \frac{\Gamma(x + 1, \mu)}{x!} \]

where \(\Gamma(x + 1, \mu)\) is the upper incomplete gamma function.

Poisson Distribution CDF

See also

Common Alternatives:

  • NegativeBinomial - The Negative Binomial is often used as an alternative the Poisson when the variance is greater than the mean (overdispersed data).

  • ZeroInflatedPoisson - The Zero-Inflated Poisson is used when there is an excess of zero counts in the data.

  • HurdlePoisson - The Hurdle Poisson is used when there is an excess of zero counts in the data.

Related Distributions:

  • Binomial - The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed. See law of rare events.

  • Normal - For sufficiently large values of \(\mu\), the normal distribution with mean \(\mu\) and standard deviation \sqrt{\mu} can be a good approximation to the Poisson.

References#