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Concept
Scale parameter
Summary
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.
If a family of probability distributions is such that there is a parameter s (and other parameters θ) for which the cumulative distribution function satisfies
then s is called a scale parameter, since its value determines the "scale" or statistical dispersion of the probability distribution. If s is large, then the distribution will be more spread out; if s is small then it will be more concentrated.
If the probability density exists for all values of the complete parameter set, then the density (as a function of the scale parameter only) satisfies
where f is the density of a standardized version of the density, i.e. .
An estimator of a scale parameter is called an estimator of scale.
In the case where a parametrized family has a location parameter, a slightly different definition is often used as follows. If we denote the location parameter by , and the scale parameter by , then we require that where is the cmd for the parametrized family. This modification is necessary in order for the standard deviation of a non-central Gaussian to be a scale parameter, since otherwise the mean would change when we rescale . However, this alternative definition is not consistently used.
We can write in terms of , as follows:
Because f is a probability density function, it integrates to unity:
By the substitution rule of integral calculus, we then have
So is also properly normalized.
Some families of distributions use a rate parameter (or "inverse scale parameter"), which is simply the reciprocal of the scale parameter. So for example the exponential distribution with scale parameter β and probability density
could equivalently be written with rate parameter λ as
The uniform distribution can be parameterized with a location parameter of and a scale parameter .
The normal distribution has two parameters: a location parameter and a scale parameter .
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