Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

9.6 LOCATION-SCALE FAMILIES

The pdf of a location-scale family of random variables maintains its basic shape when its location or scale parameters are varied.

Definition: Location-Scale Family Let X be a random variable with cdf fX(x). A location family is obtained via the transformation Y = X+u with . Its distribution has the form

(9.84)

A scale family is obtained via the transformation Y = vX, where v>0. Its distribution has the form

(9.85)

Combining the transformations to generate Y = u+vX results in a location-scale family, whose distribution has the form

(9.86)

and pdf

TABLE 9.3 Minimal Sufficient Statistics for Location-Scale Families

Statistic T	Distribution Family {Parameter }
	Cauchy , Laplace , logistic
X(1)	Shifted exponential , uniform {a, known b}
	Shifted ...