Scale parameters

The scale parameter (also scatter parameter or variability parameter ) of a probability distribution is a parameter that describes the variability (or scatter ) of the distribution; the larger the scale parameter, the broader the distribution described.

definition

For a probability distribution that is described by the (cumulative) distribution function with scale parameters , applies ${\displaystyle F(x;s)}$ ${\displaystyle s}$

${\displaystyle F\left({\frac {x}{c}};s\right)=F(x;cs),}$

d. That is, an increase in the scale parameter by the factor corresponds to a scaling of the random variable by the same factor. If a probability density function exists, then it also holds ${\displaystyle c}$ ${\displaystyle f(x;s)}$

${\displaystyle {\frac {1}{c}}f\left({\frac {x}{c}};s\right)=f(x;cs).}$

Examples

Typical scale parameters are:

• the standard deviation of the normal distribution , ${\displaystyle \sigma }$
• the parameter of the Rayleigh distribution , ${\displaystyle \sigma }$
• the reciprocal of the exponential distribution . ${\displaystyle 1/\lambda }$

literature

• Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-17260-1 , doi : 10.1007 / 978-3-642-17261-8 ( limited preview in Google book search).
• Robert Hafner: Probability and Statistics . Springer, Vienna New York 1989, ISBN 978-3-211-82162-6 , doi : 10.1007 / 978-3-7091-6944-5 ( limited preview in Google book search).