location–scale family

{{Short description|Family of probability distributions}}

In probability theory, especially in mathematical statistics, a location–scale family is a family of probability distributions parametrized by a location parameter and a non-negative scale parameter. For any random variable $X$ whose probability distribution function belongs to such a family, the distribution function of $Y\; \backslash stackrel\{d\}\{=\}\; a\; +\; b\; X$also belongs to the family (where $\backslash stackrel\{d\}\{=\}$ means "equal in distribution"—that is, "has the same distribution as").

In other words, a class $\backslash Omega$ of probability distributions is a location–scale family if for all cumulative distribution functions $F\; \backslash in\; \backslash Omega$ and any real numbers $a\; \backslash in\; \backslash mathbb\{R\}$ and $b\; >\; 0$, the distribution function $G(x)\; =\; F(a\; +\; b\; x)$ is also a member of $\backslash Omega$.

• If $X$ has a cumulative distribution function $F\_X(x)=\; P(X\backslash le\; x)$, then $Y\{=\}\; a\; +\; b\; X$ has a cumulative distribution function $F\_Y(y)\; =\; F\_X\backslash left(\backslash frac\{y-a\}\{b\}\backslash right)$.
• If $X$ is a discrete random variable with probability mass function $p\_X(x)=\; P(X=x)$, then $Y\{=\}\; a\; +\; b\; X$ is a discrete random variable with probability mass function $p\_Y(y)\; =\; p\_X\backslash left(\backslash frac\{y-a\}\{b\}\backslash right)$.
• If $X$ is a continuous random variable with probability density function $f\_X(x)$, then $Y\{=\}\; a\; +\; b\; X$ is a continuous random variable with probability density function $f\_Y(y)\; =\; \backslash frac\{1\}\{b\}f\_X\backslash left(\backslash frac\{y-a\}\{b\}\backslash right)$.
• If $X$ has a moment generating function $M\_X(t)$, then $Y\{=\}\; a\; +\; b\; X$ has a moment generating function $M\_Y(t)=e^\{at\}M\_X(bt)$.
• If $X$ has a characteristic function $\backslash varphi\_X(t)$, then $Y\{=\}\; a\; +\; b\; X$ has a characteristic function $\backslash varphi\_Y(t)=e^\{iat\}\backslash varphi\_X(bt)$.

Moreover, if $X$ and $Y$ are two random variables whose distribution functions are members of the family, and assuming existence of the first two moments and $X$ has zero mean and unit variance,

then $Y$ can be written as $Y\; \backslash stackrel\{d\}\{=\}\; \backslash mu\_Y\; +\; \backslash sigma\_Y\; X$ , where $\backslash mu\_Y$ and $\backslash sigma\_Y$ are the mean and standard deviation of $Y$.

In decision theory, if all alternative distributions available to a decision-maker are in the same location–scale family, and the first two moments are finite, then a two-moment decision model can apply, and decision-making can be framed in terms of the means and the variances of the distributions.{{cite journal |last=Meyer |first=Jack |title=Two-Moment Decision Models and Expected Utility Maximization |journal=American Economic Review |volume=77 |year=1987 |issue=3 |pages=421–430 |jstor=1804104 }}{{cite journal |last=Mayshar |first=J. |title=A Note on Feldstein's Criticism of Mean-Variance Analysis |journal=Review of Economic Studies |volume=45 |issue=1 |year=1978 |pages=197–199 |jstor=2297094 }}{{cite book |last=Sinn |first=H.-W. |author-link=Hans-Werner Sinn |title=Economic Decisions under Uncertainty |edition=Second English |year=1983 |publisher=North-Holland }}