In probability theory, especially as that field is used in statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter and a non-negative scale parameter; if X is any random variable whose probability distribution belongs to such a family, then Y =d (a + bX) is another (where =d means "is equal in distribution to" — that is, "has the same distribution as"), and every distribution in the family is of that form. Moreover, if X is a zero-mean, unit-variance member of the family, then every member Y of the family can be written as Y =d (μY + σYX), where μY and σY are the mean and standard deviation of Y.
In other words, a class Ω of probability distributions is a location-scale family if whenever F is the cumulative distribution function of a member of Ω and a is any real number and b > 0, then G(x) = F(a + bx) is also the cumulative distribution function of a member of Ω.
In decision theory, if all alternative distributions available to a decision-maker are in the same location-scale family, then a two-moment decision model applies, and decision-making can be framed in terms of the means and the variances of the distributions.
Read more about Location-scale Family: Examples, Converting A Single Distribution To A Location-scale Family
Famous quotes containing the word family:
“In the capsule biography by which most of the people knew one another, I was understood to be an Air Force pilot whose family was wealthy and lived in the East, and I even added the detail that I had a broken marriage and drank to get over it.... I sometimes believed what I said and tried to take the cure in the very real sun of Desert D’Or with its cactus, its mountain, and the bright green foliage of its love and its money.”
—Norman Mailer (b. 1923)