# Scale parameter

A so-called positional parameter, which parametrizes a family of probability distributions of one type. A distribution in $ \mathbf R $
with distribution function $ F $
is said to belong to the same type as a fixed distribution with distribution function $ F _ {0} $
if $ F ( x) = F _ {0} (( x - b)/a) $.
Here $ a > 0 $
is the scale parameter and $ b $
is the shift parameter (or centralizing parameter). The meaning of the scale parameter is as follows: If $ F _ {0} $
and $ F $
are the distribution functions of random variables $ X _ {0} $
and $ X $,
respectively, then a transition from $ X _ {0} $
to $ X = aX _ {0} $(
for $ b = 0 $)
represents a change in the unit of measurement.

#### Comments

The (possibly multi-dimensional) parameter $ b $ in the family $ F _ {0} ( ( x- b) / a ) $ is also called the location parameter. The whole family of distributions is sometimes called a location-scale family of distributions.

#### References

[a1] | L. Breiman, "Statistics with a view towards applications" , Houghton Mifflin (1973) pp. 34–40 |

**How to Cite This Entry:**

Scale parameter.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Scale_parameter&oldid=48615