===========================================
Distribution Properties
===========================================


.. _continuous:

Continuous
---------

These distributions are defined over a continuous range of values.

- `Asymmetric Laplace <./distributions/gallery/asymmetric_laplace.html>`_
- `Beta <./distributions/gallery/beta.html>`_
- `Beta Scaled <./distributions/gallery/beta_scaled.html>`_
- `Cauchy <./distributions/gallery/cauchy.html>`_
- `Chi-Squared <./distributions/gallery/chisquared.html>`_
- `Ex-Gaussian <./distributions/gallery/exgaussian.html>`_
- `Exponential <./distributions/gallery/exponential.html>`_
- `Gamma <./distributions/gallery/gamma.html>`_
- `Gumbel <./distributions/gallery/gumbel.html>`_
- `Half-Cauchy <./distributions/gallery/halfcauchy.html>`_
- `Half-Normal <./distributions/gallery/halfnormal.html>`_
- `Half-Student's t <./distributions/gallery/halfstudentt.html>`_
- `Inverse Gamma <./distributions/gallery/inversegamma.html>`_
- `Kumaraswamy <./distributions/gallery/kumaraswamy.html>`_
- `Laplace <./distributions/gallery/laplace.html>`_
- `Log-Normal <./distributions/gallery/log_normal.html>`_
- `Logistic <./distributions/gallery/logistic.html>`_
- `Log-Logistic <./distributions/gallery/log_logistic.html>`_
- `Logit-Normal <./distributions/gallery/logit_normal.html>`_
- `Moyal <./distributions/gallery/moyal.html>`_
- `Normal <./distributions/gallery/normal.html>`_
- `Pareto <./distributions/gallery/pareto.html>`_
- `Rice <./distributions/gallery/rice.html>`_
- `Scaled Inverse Chi-Squared <./distributions/gallery/scaledinversechisquared.html>`_
- `Skew-Normal <./distributions/gallery/skewnormal.html>`_
- `Student's t <./distributions/gallery/students_t.html>`_
- `Skew-Student's t <./distributions/gallery/skew_studentt.html>`_
- `Triangular <./distributions/gallery/triangular.html>`_
- `Truncated Normal <./distributions/gallery/truncated_normal.html>`_
- `Uniform <./distributions/gallery/uniform.html>`_
- `Von Mises <./distributions/gallery/vonmises.html>`_
- `Wald <./distributions/gallery/wald.html>`_
- `Weibull <./distributions/gallery/weibull.html>`_
- `Dirichlet <./distributions/gallery/dirichlet.html>`_
- `Multivariate Normal <./distributions/gallery/mvnormal.html>`_

.. _discrete:

Discrete
--------

These distributions are defined over a discrete range of values.

- `Bernoulli <./distributions/gallery/bernoulli.html>`_
- `Beta Binomial <./distributions/gallery/betabinomial.html>`_
- `Binomial <./distributions/gallery/binomial.html>`_
- `Categorical <./distributions/gallery/categorical.html>`_
- `Discrete Uniform <./distributions/gallery/discrete_uniform.html>`_
- `Discrete Weibull <./distributions/gallery/discrete_weibull.html>`_
- `Geometric <./distributions/gallery/geometric.html>`_
- `Hypergeometric <./distributions/gallery/hypergeometric.html>`_
- `Negative Binomial <./distributions/gallery/negativebinomial.html>`_
- `Poisson <./distributions/gallery/poisson.html>`_
- `Zero-Inflated Binomial <./distributions/gallery/zeroinflatedbinomial.html>`_
- `Zero-Inflated Negative Binomial <./distributions/gallery/zeroinflatednegativebinomial.html>`_
- `Zero-Inflated Poisson <./distributions/gallery/zeroinflatedpoisson.html>`_

.. _symmetric:

Symmetric
---------
These distributions are symmetric about their center (or can be symmetric under parameter choices):

- `Laplace <./distributions/gallery/laplace.html>`_
- `Normal <./distributions/gallery/normal.html>`_
- `Cauchy <./distributions/gallery/cauchy.html>`_
- `Student's t <./distributions/gallery/students_t.html>`_
- `Beta (when α = β) <./distributions/gallery/beta.html>`_
- `Beta Scaled (when α = β) <./distributions/gallery/beta_scaled.html>`_
- `Uniform <./distributions/gallery/uniform.html>`_
- `Logistic <./distributions/gallery/logistic.html>`_
- `Von Mises <./distributions/gallery/vonmises.html>`_ *(Note: circular distributions such as Von Mises are symmetric on the circle.)*
- `Multivariate Normal <./distributions/gallery/mvnormal.html>`_

*Note: PreliZ distributions have a `skewness()` method.*  

For more information on symmetric distributions, see:  
`Symmetric distribution  <https://en.wikipedia.org/wiki/Symmetric_distribution>`_.

.. _asymmetric:

Asymmetric
----------
Distributions with skewed (non‐symmetric) shapes include:

- `Asymmetric Laplace <./distributions/gallery/asymmetric_laplace.html>`_
- `Exponential <./distributions/gallery/exponential.html>`_
- `Gamma <./distributions/gallery/gamma.html>`_
- `Gumbel <./distributions/gallery/gumbel.html>`_
- `Inverse Gamma <./distributions/gallery/inversegamma.html>`_
- `Kumaraswamy <./distributions/gallery/kumaraswamy.html>`_
- `Log-Logistic <./distributions/gallery/log_logistic.html>`_
- `Logit-Normal <./distributions/gallery/logit_normal.html>`_
- `Ex-Gaussian <./distributions/gallery/exgaussian.html>`_
- `Moyal <./distributions/gallery/moyal.html>`_
- `Pareto <./distributions/gallery/pareto.html>`_
- `Rice <./distributions/gallery/rice.html>`_
- `Scaled Inverse Chi-Squared <./distributions/gallery/scaledinversechisquared.html>`_
- `Skew-Normal <./distributions/gallery/skewnormal.html>`_
- `Skew-Student's t <./distributions/gallery/skew_studentt.html>`_
- `Weibull <./distributions/gallery/weibull.html>`_
- `Geometric <./distributions/gallery/geometric.html>`_
- `Wald <./distributions/gallery/wald.html>`_

*Note: PreliZ distributions have a `skewness()` method.*  

For more on skewness and asymmetry, see:  
`Skewness  <https://en.wikipedia.org/wiki/Skewness>`_.

.. _bounded:

Bounded
-------
Distributions whose support is bounded (i.e. defined only on a finite interval).
**Continuous:**
- `Beta <./distributions/gallery/beta.html>`_
- `Beta Scaled <./distributions/gallery/beta_scaled.html>`_
- `Triangular <./distributions/gallery/triangular.html>`_
- `Uniform <./distributions/gallery/uniform.html>`_
- `Logit-Normal <./distributions/gallery/logit_normal.html>`_
- `Kumaraswamy <./distributions/gallery/kumaraswamy.html>`_
- `Truncated Normal <./distributions/gallery/truncated_normal.html>`_
- `Dirichlet <./distributions/gallery/dirichlet.html>`_

**Discrete:**
- `Beta Binomial <./distributions/gallery/betabinomial.html>`_
- `Binomial <./distributions/gallery/binomial.html>`_
- `Categorical <./distributions/gallery/categorical.html>`_
- `Discrete Uniform <./distributions/gallery/discrete_uniform.html>`_
- `Hypergeometric <./distributions/gallery/hypergeometric.html>`_
- `Zero-Inflated Binomial <./distributions/gallery/zeroinflatedbinomial.html>`_

For more on distributions with bounded support, see:  
`Support (statistics) <https://en.wikipedia.org/wiki/Support_(statistics)>`_.

.. _unbounded:

Unbounded
---------
Distributions defined over the entire real line (or “two‐sided” with infinite support):

- `Asymmetric Laplace <./distributions/gallery/asymmetric_laplace.html>`_
- `Ex-Gaussian <./distributions/gallery/exgaussian.html>`_
- `Gumbel <./distributions/gallery/gumbel.html>`_
- `Normal <./distributions/gallery/normal.html>`_
- `Cauchy <./distributions/gallery/cauchy.html>`_
- `Student's t <./distributions/gallery/students_t.html>`_
- `Laplace <./distributions/gallery/laplace.html>`_
- `Logistic <./distributions/gallery/logistic.html>`_
- `Moyal <./distributions/gallery/moyal.html>`_
- `Skew-Normal <./distributions/gallery/skewnormal.html>`_
- `Skew-Student's t <./distributions/gallery/skew_studentt.html>`_
- `Multivariate Normal <./distributions/gallery/mvnormal.html>`_

For more on distributions with unbounded support, see:  
`Support (statistics) <https://en.wikipedia.org/wiki/Support_(statistics)>`_.

.. _non_negative:

Non‐Negative
------------
Distributions supported on the non‐negative real numbers.

**Continuous:**
- `Exponential <./distributions/gallery/exponential.html>`_
- `Gamma <./distributions/gallery/gamma.html>`_
- `Chi-Squared <./distributions/gallery/chisquared.html>`_
- `Weibull <./distributions/gallery/weibull.html>`_
- `Pareto <./distributions/gallery/pareto.html>`_
- `Half-Cauchy <./distributions/gallery/halfcauchy.html>`_
- `Half-Normal <./distributions/gallery/halfnormal.html>`_
- `Half-Student's t <./distributions/gallery/halfstudentt.html>`_
- `Inverse Gamma <./distributions/gallery/inversegamma.html>`_
- `Log-Normal <./distributions/gallery/log_normal.html>`_
- `Rice <./distributions/gallery/rice.html>`_
- `Wald <./distributions/gallery/wald.html>`_
- `Log-Logistic <./distributions/gallery/log_logistic.html>`_
- `Logit-Normal <./distributions/gallery/logit_normal.html>`_
- `Scaled Inverse Chi-Squared <./distributions/gallery/scaledinversechisquared.html>`_


**Discrete:**
- `Bernoulli <./distributions/gallery/bernoulli.html>`_
- `Binomial <./distributions/gallery/binomial.html>`_
- `Negative Binomial <./distributions/gallery/negativebinomial.html>`_
- `Poisson <./distributions/gallery/poisson.html>`_
- `Zero-Inflated Binomial <./distributions/gallery/zeroinflatedbinomial.html>`_
- `Zero-Inflated Negative Binomial <./distributions/gallery/zeroinflatednegativebinomial.html>`_
- `Zero-Inflated Poisson <./distributions/gallery/zeroinflatedpoisson.html>`_
- `Discrete Weibull <./distributions/gallery/discrete_weibull.html>`_
- `Geometric <./distributions/gallery/geometric.html>`_

For more on non‐negative random variables and distributions, see:  
`Support (statistics) <https://en.wikipedia.org/wiki/Support_(statistics)>`_.

.. _multivariate:

Multivariate
------------
Distributions with more than one dimension:

- `Dirichlet <./distributions/gallery/dirichlet.html>`_
- `Multivariate Normal <./distributions/gallery/mvnormal.html>`_

For more on multivariate probability distributions, see:  
`Joint probability distribution  <https://en.wikipedia.org/wiki/Joint_probability_distribution>`_.

.. _univariate:

Univariate
----------
The following distributions are univariate (one-dimensional).

**Continuous:**
- `Asymmetric Laplace <./distributions/gallery/asymmetric_laplace.html>`_
- `Beta <./distributions/gallery/beta.html>`_
- `Beta Scaled <./distributions/gallery/beta_scaled.html>`_
- `Cauchy <./distributions/gallery/cauchy.html>`_
- `Chi-Squared <./distributions/gallery/chisquared.html>`_
- `Ex-Gaussian <./distributions/gallery/exgaussian.html>`_
- `Exponential <./distributions/gallery/exponential.html>`_
- `Gamma <./distributions/gallery/gamma.html>`_
- `Gumbel <./distributions/gallery/gumbel.html>`_
- `Half-Cauchy <./distributions/gallery/halfcauchy.html>`_
- `Half-Normal <./distributions/gallery/halfnormal.html>`_
- `Half-Student's t <./distributions/gallery/halfstudentt.html>`_
- `Inverse Gamma <./distributions/gallery/inversegamma.html>`_
- `Kumaraswamy <./distributions/gallery/kumaraswamy.html>`_
- `Laplace <./distributions/gallery/laplace.html>`_
- `Log-Normal <./distributions/gallery/log_normal.html>`_
- `Logistic <./distributions/gallery/logistic.html>`_
- `Log-Logistic <./distributions/gallery/log_logistic.html>`_
- `Logit-Normal <./distributions/gallery/logit_normal.html>`_
- `Moyal <./distributions/gallery/moyal.html>`_
- `Normal <./distributions/gallery/normal.html>`_
- `Pareto <./distributions/gallery/pareto.html>`_
- `Rice <./distributions/gallery/rice.html>`_
- `Scaled Inverse Chi-Squared <./distributions/gallery/scaledinversechisquared.html>`_
- `Skew-Normal <./distributions/gallery/skewnormal.html>`_
- `Student's t <./distributions/gallery/students_t.html>`_
- `Skew-Student's t <./distributions/gallery/skew_studentt.html>`_
- `Triangular <./distributions/gallery/triangular.html>`_
- `Truncated Normal <./distributions/gallery/truncated_normal.html>`_
- `Uniform <./distributions/gallery/uniform.html>`_
- `Von Mises <./distributions/gallery/vonmises.html>`_
- `Wald <./distributions/gallery/wald.html>`_
- `Weibull <./distributions/gallery/weibull.html>`_

**Discrete:**
- `Bernoulli <./distributions/gallery/bernoulli.html>`_
- `Beta Binomial <./distributions/gallery/betabinomial.html>`_
- `Binomial <./distributions/gallery/binomial.html>`_
- `Categorical <./distributions/gallery/categorical.html>`_
- `Discrete Uniform <./distributions/gallery/discrete_uniform.html>`_
- `Discrete Weibull <./distributions/gallery/discrete_weibull.html>`_
- `Geometric <./distributions/gallery/geometric.html>`_
- `Hypergeometric <./distributions/gallery/hypergeometric.html>`_
- `Negative Binomial <./distributions/gallery/negativebinomial.html>`_
- `Poisson <./distributions/gallery/poisson.html>`_
- `Zero-Inflated Binomial <./distributions/gallery/zeroinflatedbinomial.html>`_
- `Zero-Inflated Negative Binomial <./distributions/gallery/zeroinflatednegativebinomial.html>`_
- `Zero-Inflated Poisson <./distributions/gallery/zeroinflatedpoisson.html>`_

For more on univariate probability distributions, see:  
`Probability distribution  <https://en.wikipedia.org/wiki/Probability_distribution>`_.

.. _modifiers:

Modifiers (Special Cases)
-------------------------
- `Censored <./distributions/gallery/censored.html>`_
- `Hurdle <./distributions/gallery/hurdle.html>`_
- `Mixture <./distributions/gallery/mixture.html>`_
- `Truncated <./distributions/gallery/truncated.html>`_

For more details on distribution modifications, see:  
`Censoring (statistics) <https://en.wikipedia.org/wiki/Censoring_(statistics)>`_,  
`Hurdle model  <https://en.wikipedia.org/wiki/Hurdle_model>`_,  
`Mixture model  <https://en.wikipedia.org/wiki/Mixture_model>`_, and  
`Truncated distribution  <https://en.wikipedia.org/wiki/Truncated_distribution>`_.

.. _heavy_tailed:

Heavy-Tailed
------------
Distributions with tails that decay slowly (i.e. they allow for large outliers) include:

- `Cauchy <./distributions/gallery/cauchy.html>`_
- `Student's t <./distributions/gallery/students_t.html>`_
- `Pareto <./distributions/gallery/pareto.html>`_
- `Half-Cauchy <./distributions/gallery/halfcauchy.html>`_
- `Half-Student's t <./distributions/gallery/halfstudentt.html>`_
- `Inverse Gamma <./distributions/gallery/inversegamma.html>`_
- `Log-Normal <./distributions/gallery/log_normal.html>`_
- `Log-Logistic <./distributions/gallery/log_logistic.html>`_
- `Skew-Student's t <./distributions/gallery/skew_studentt.html>`_

*Note: PreliZ distributions have a `kurtosis()` method.*  

For more on heavy-tailed distributions, see:  
`Heavy-tailed distribution  <https://en.wikipedia.org/wiki/Heavy-tailed_distribution>`_.

.. _light_tailed:

Light-Tailed
------------
Distributions with tails that decay relatively quickly include:

- `Asymmetric Laplace <./distributions/gallery/asymmetric_laplace.html>`_
- `Chi-Squared <./distributions/gallery/chisquared.html>`_
- `Ex-Gaussian <./distributions/gallery/exgaussian.html>`_
- `Exponential <./distributions/gallery/exponential.html>`_
- `Gamma <./distributions/gallery/gamma.html>`_
- `Half-Normal <./distributions/gallery/halfnormal.html>`_
- `Laplace <./distributions/gallery/laplace.html>`_
- `Logistic <./distributions/gallery/logistic.html>`_
- `Moyal <./distributions/gallery/moyal.html>`_
- `Normal <./distributions/gallery/normal.html>`_
- `Rice <./distributions/gallery/rice.html>`_
- `Skew-Normal <./distributions/gallery/skewnormal.html>`_
- `Truncated Normal <./distributions/gallery/truncated_normal.html>`_
- `Triangular <./distributions/gallery/triangular.html>`_
- `Wald <./distributions/gallery/wald.html>`_
- `Weibull <./distributions/gallery/weibull.html>`_

*Note: PreliZ distributions have a `kurtosis()` method.*  

For more on tail behavior and light-tailed distributions, see:  
`Heavy-tailed distribution  <https://en.wikipedia.org/wiki/Heavy-tailed_distribution>`_.

.. _extreme_value:

Extreme Value
-------------
Distributions commonly used in the modeling of extreme events:

- `Gumbel <./distributions/gallery/gumbel.html>`_
- `Log-Logistic <./distributions/gallery/log_logistic.html>`_

For more on extreme value theory, see:  
`Extreme value theory  <https://en.wikipedia.org/wiki/Extreme_value_theory>`_.

.. _zero_inflated:

Zero-Inflated
-------------
These distributions have been augmented to allow for extra zeros:

- `Zero-Inflated Poisson <./distributions/gallery/zeroinflatedpoisson.html>`_
- `Zero-Inflated Binomial <./distributions/gallery/zeroinflatedbinomial.html>`_
- `Zero-Inflated Negative Binomial <./distributions/gallery/zeroinflatednegativebinomial.html>`_

For more on zero-inflated models, see:  
`Zero-inflated model  <https://en.wikipedia.org/wiki/Zero-inflated_model>`_.