In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, a shape parameter is a kind of numerical parameter of a parametric family of probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

s.

Definition

A shape parameter is any parameter of a probability distribution that is neither a location parameter

Location parameter

In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...

 nor a scale parameter

Scale parameter

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...

 (nor a function of either or both of these only, such as a rate parameter). Such a parameter must affect the shape of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does).

Examples

The following continuous probability distributions have a shape parameter:

• Beta distribution
• Burr distribution
• Erlang distribution
• Exponential power distribution
• Gamma distribution
• Generalized extreme value distribution
• Log-logistic distribution
  Log-logistic distribution
  In probability and statistics, the log-logistic distribution is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, for example mortality from cancer following...
  
  
• Inverse-gamma distribution
• Pareto distribution
• Pearson distribution
  Pearson distribution
  The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.- History :...
  
  
• Tukey lambda distribution
• Weibull distribution
• Student's t-distribution

By contrast, the following continuous distributions do not have a shape parameter, so their shape is fixed and only their location or their scale or both can change. It follows that (where they exist) the skewness

Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...

 and kurtosis

Kurtosis

In probability theory and statistics, kurtosis is any measure of the "peakedness" of the probability distribution of a real-valued random variable...

 of these distribution are constants, as skewness and kurtosis are independent of location and scale parameters.

• Exponential distribution
  Exponential distribution
  In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
  
  
• Cauchy distribution
  Cauchy distribution
  The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...
  
  
• Logistic distribution
• Normal distribution
• Raised cosine distribution
• Uniform distribution
  Uniform distribution
  -Probability theory:* Discrete uniform distribution* Continuous uniform distribution-Other:* "Uniform distribution modulo 1", see Equidistributed sequence*Uniform distribution , a type of species distribution* Distribution of military uniforms...
  
  
• Wigner semicircle distribution