Shape parameter
In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.
Definition
A shape parameter is any parameter of a probability distribution that is neither a location parameter nor a scale parameter (nor a function of either of both or 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
- Inverse-gamma distribution
- Pareto distribution
- Pearson distribution
- Weibull 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:
- Exponential distribution
- Cauchy distribution
- Logistic distribution
- Normal distribution
- Raised cosine distribution
- Uniform distribution
- Wigner semicircle distribution
See also
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