Editing Scale parameter (section)

==Definition==
If a family of [[probability distribution]]s is such that there is a parameter ''s'' (and other parameters ''θ'') for which the [[cumulative distribution function]] satisfies

:<math>F(x;s,\theta) = F(x/s;1,\theta), \!</math>

then ''s'' is called a '''scale parameter''', since its value determines the "[[scale (ratio)|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.

[[File:Effects of a scale parameter on a positive-support probability distribution.gif|thumb|300px|Animation showing the effects of a scale parameter on a probability distribution supported on the positive real line.]]
[[File:Effect of a scale parameter over a mixture of two normal probability distributions.gif|thumb|300px|Effect of a scale parameter over a mixture of two normal probability distributions]]

If the [[probability density function|probability density]] exists for all values of the complete parameter set, then the density (as a function of the scale parameter only) satisfies
:<math>f_s(x) = f(x/s)/s, \!</math>
where ''f'' is the density of a standardized version of the density, i.e. <math>f(x) \equiv f_{s=1}(x)</math>.

An [[estimator]] of a scale parameter is called an '''estimator of scale.'''

===Families with Location Parameters===
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 <math>m</math>, and the scale parameter by <math>s</math>, then we require that <math>F(x;s,m,\theta)=F((x-m)/s;1,0,\theta)</math> where <math>F(x,s,m,\theta)</math> is the CDF for the parametrized family.<ref>{{cite web |url= http://www.encyclopediaofmath.org/index.php?title=Scale_parameter&oldid=13206 |title= Scale parameter
 |last=Prokhorov |first=A.V. |date= 7 February 2011 |website=Encyclopedia of Mathematics |publisher= Springer |access-date=7 February 2019}}</ref> 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 <math>x</math>. However, this alternative definition is not consistently used.<ref>{{cite web |url=https://www.math.kth.se/matstat/gru/sf2955/scaleparameter |title= Scale parameter
 |last=Koski |first=Timo |website=KTH Royal Institute of Technology|access-date=7 February 2019}}</ref>

===Simple manipulations===
We can write <math>f_s</math> in terms of <math>g(x) = x/s</math>, as follows:

:<math>f_s(x) = f\left(\frac{x}{s}\right) \cdot \frac{1}{s} = f(g(x))g'(x).</math>

Because ''f'' is a probability density function, it integrates to unity:

:<math>
 1 = \int_{-\infty}^{\infty} f(x)\,dx
   = \int_{g(-\infty)}^{g(\infty)} f(x)\,dx.
 </math>

By the [[substitution rule]] of integral calculus, we then have

:<math>
 1 = \int_{-\infty}^{\infty} f(g(x)) g'(x)\,dx
   = \int_{-\infty}^{\infty} f_s(x)\,dx.
 </math>

So <math>f_s</math> is also properly normalized.