Distributions

General Information:

• NIST Gallery of Distributions

• StatSoft index of distributions

Other resources (please verify these yourself before using):

• http://statpages.org/javastat.html#Tables

The following content is a partial listing of some of the more commonly-recognized probability distributions, both discrete and continuous.

DISCRETE DISTRIBUTIONS

Hypergeometric Distribution

General

More information

• NIST - Hypergeometric Distribution

• StatSoft - Hypergeometric Distribution (n/a)

• Wikipedia - Hypergeometric Distribution

• Arsham - Hypergeometric Distribution

Binomial Distribution

General

• Bernoulli Trial (StatSoft)

More Information

• NIST - Binomial Distribution

• StatSoft - Binomial Distribution

• Wikipedia - Binomial Distribution

• Arsham - Binomial Distribution

Negative Binomial Distribution

General

More Information

• NIST - Negative Binomial Distribution (n/a)

• StatSoft - Negative Binomial Distribution (n/a)

• Wikipedia - Negative Binomial Distribution

• Arsham - Negative Binomial Distribution

See also:

• Pascal distribution

• Polya distribution

Geometric Distribution

General

More Information

• NIST - Geometric Distribution (n/a)

• StatSoft - Geometric Distribution

• Wikipedia - Geometric Distribution

• Arsham - Geometric Distribution

Poisson Distribution

General

More Information

• NIST - Poisson Distribution

• StatSoft - Poisson Distribution

• Wikipedia - Poisson Distribution

• Arsham - Poisson Distribution

Uniform Distribution (discrete)

General

More Information

• NIST - Uniform Distribution (discrete)

• StatSoft - Uniform Distribution (discrete)

• Wikipedia - Uniform Distribution (discrete)

• Arsham - Uniform Distribution (discrete)

CONTINUOUS DISTRIBUTIONS

Normal Distribution

Other names: Gaussian Distribution

General

More Information

• NIST - Normal Distribution

• StatSoft - Normal Distribution

• Wikipedia - Normal Distribution

• Arsham - Normal Distribution

See also:

• Multivariate Normal Distribution [Wikipedia]

Log-normal Distribution

General

More Information

• NIST - Lognormal Distribution

• StatSoft - Lognormal Distribution

• Wikipedia - Lognormal Distribution

• Arsham - Lognormal Distribution

Uniform Distribution (continuous)

Other names: Rectangular Distribution

General

• Related to the Beta distribution.

More Information

• NIST - Uniform Distribution

• StatSoft - Uniform Distribution

• Wikipedia - Uniform Distribution

• Arsham - Uniform Distribution

Chi-square Distribution

General

• Related to the Standard Normal (Z) and Gamma distributions.

More Information

• NIST - Chi-square Distribution

• StatSoft - Chi-square Distribution

• Wikipedia - Chi-square Distribution

• Arsham - Chi-square Distribution

Gamma Distribution

General

• Related to the Chi-square distribution.

More Information

• NIST - Gamma Distribution

• StatSoft - Gamma Distribution

• Wikipedia - Gamma Distribution

• Arsham - Gamma Distribution

Beta Distribution

General

More Information

• NIST - Beta Distribution

• StatSoft - Beta Distribution

• Wikipedia - Beta Distribution

• Arsham - Beta Distribution

Exponential Distribution

General

More Information

• NIST - Exponential Distribution

• StatSoft - Exponential Distribution

• Wikipedia - Exponential Distribution

• Arsham - Exponential Distribution

Weibull Distribution

General

More Information

• NIST - Weibull Distribution

• StatSoft - Weibull Distribution

• Wikipedia - Weibull Distribution

• Arsham - Weibull Distribution

Notes from the R package "survival" ('survreg' function):

• There are multiple ways to parameterize a Weibull distribution. The survreg function imbeds it in a general location-scale familiy, which is a different parameterization than the rweibull function, and often leads to confusion.

• survreg's scale = 1/(rweibull shape)

• survreg's intercept = log(rweibull scale)

• For the log-likelihood all parameterizations lead to the same value.

The same issues are found in SAS, depending on which procedure is used. Anyone using software to estimate weibull distribution parameters should be aware of this issue.

Logistic Distribution

General

More Information

• NIST - Logistic Distribution

• StatSoft - Logistic Distribution

• Wikipedia - Logistic Distribution

• Arsham - Logistic Distribution

F Distribution

General

More Information

• NIST - F Distribution

• StatSoft - F Distribution

• Wikipedia - F Distribution

• Arsham - F Distribution

Student's t Distribution

General

More Information

• NIST - Student's t Distribution

• StatSoft - Student's t Distribution

• Wikipedia - Student's t Distribution

• Arsham - Student's t Distribution

R code for graphically comparing the Normal and Student's t distributions is in the file "Compare Normal vs t dist.R" at the bottom of this page.

(Courtesy of Dr. Juan Hu)

Standard Normal (Z) Distribution

General

• Related to the Chi-square distribution.

More Information

• NIST - Z Distribution

• StatSoft - Z Distribution (n/a)

• Wikipedia - Z Distribution

• Arsham - Z Distribution

Other interesting distributions

Cauchy Distribution

General Information

• An interesting aspect of the Cauchy distribution is that its variance (second moment about the mean) is undefined.

See also:

• NIST - Cauchy Distribution

• StatSoft - Cauchy Distribution

• Wikipedia - Cauchy Distribution

• Arsham - Cauchy Distribution

Rayleigh Distribution

General

More Information

• NIST - Rayleigh Distribution

• StatSoft - Rayleigh Distribution

• Wikipedia - Rayleigh Distribution

• Arsham - Rayleigh Distribution

Extreme Value Distribution

General

More Information

• NIST - Extreme Value Distribution

• StatSoft - Extreme Value Distribution

• Wikipedia - Extreme Value Distribution

• Arsham - Extreme Value Distribution

Double Exponential (aka Laplace) Distribution

General

More Information

• NIST - Double Exponential (Laplace) Distribution

• StatSoft - Double Exponential (Laplace) Distribution

• Wikipedia - Double Exponential (Laplace) Distribution

• Arsham - Double Exponential (Laplace) Distribution

Generalized Gamma Distribution

General

More Information

• NIST - Generalized Gamma Distribution

• StatSoft - Generalized Gamma Distribution

• Wikipedia - Generalized Gamma Distribution

• Arsham - Generalized Gamma Distribution

• Weibull.com - Generalized Gamma Distribution

Gompertz Distribution

General

More Information

• NIST - Gompertz Distribution

• StatSoft - Gompertz Distribution

• Wikipedia - Gompertz Distribution

• Arsham - Gompertz Distribution

Pareto Distribution

General

More Information

• NIST - Pareto Distribution

• StatSoft - Pareto Distribution

• Wikipedia - Pareto Distribution

• Arsham - Pareto Distribution

Triangular Distribution

General

More Information

• NIST - Triangular Distribution

• StatSoft - Triangular Distribution (n/a)

• Wikipedia - Triangular Distribution

• Arsham - Triangular Distribution