TypeOfADistributionFunction 2006-11-22 22:50:34 2006-11-29 23:39:20 Definition distribution function type scale factor location factor standard distribution function location family scale family \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} \usepackage{pst-plot} \usepackage{psfrag} % define commands here Two distribution functions $F,G:\mathbb{R}\to [0,1]$ are said to of the same \emph{type} if there exist $a,b\in\mathbb{R}$ such that $G(x)=F(ax+b)$. $a$ is called the \emph{scale parameter}, and $b$ the \emph{location parameter} or \emph{centering parameter}. Let's write $F\stackrel{t}{=}G$ to denote that $F$ and $G$ are of the same type. \textbf{Remarks}. \begin{itemize} \item Necessarily $a>0$, for otherwise at least one of $G(-\infty)=0$ or $G(\infty)=1$ would be violated. \item If $G(x)=F(x+b)$, then the graph of $G$ is \emph{shifted} to the right from the graph of $F$ by $b$ units, if $b>0$ and to the left if $b<0$. \item If $G(x)=F(ax)$, then the graph of $G$ is \emph{stretched} from the graph of $F$ by $a$ units if $a>1$, and \emph{compressed} if $a<1$. \item If $X$ and $Y$ are random variables whose distribution functions are of the same type, say, $F$ and $G$ respectively, and related by $G(x)=F(ax+b)$, then $X$ and $aY+b$ are identically distributed, since $$P(X\le z)=F(z)=G(\frac{z-b}{a})=P(Y \le \frac{z-b}{a})=P(aY+b \le z).$$ When $X$ and $aY+b$ are identically distributed, we write $X \stackrel{t}{=} Y$. \item Again, suppose $X$ and $Y$ correspond to $F$ and $G$, two distribution functions of the same type related by $G(x)=F(ax+b)$. Then it is easy to see that $E[X]<\infty$ iff $E[Y]<\infty$. In fact, if the expectation exists for one, then $E[X]=aE[Y]+b$. Furthermore, $Var[X]$ is finite iff $Var[Y]$ is. And in this case, $Var[X]=a^2Var[Y]$. In general, convergence of moments is a ``typical'' property. \item We can partition the set of distribution functions into disjoint subsets of functions belonging to the same types, since the binary relation $\stackrel{t}{=}$ is an equivalence relation. \item By the same token, we can classify all real random variables defined on a fixed probability space according to their distribution functions, so that if $X$ and $Y$ are of the same type $\tau$ iff their corresponding distribution functions $F$ and $G$ are of type $\tau$. \item Given an equivalence class of distribution functions belonging to a certain type $\tau$, such that a random variable $Y$ of type $\tau$ exists with finite expectation and variance, then there is one distribution function $F$ of type $\tau$ corresponding to a random variable $X$ such that $E[X]=0$ and $Var[X]=1$. $F$ is called the \emph{standard distribution function} for type $\tau$. For example, the standard (cumulative) normal distribution is the standard distribution function for the type consisting of all normal distribution functions. \item Within each type $\tau$, we can further classify the distribution functions: if $G(x)=F(x+b)$, then we say that $G$ and $F$ belong to the same \emph{location family} under $\tau$; and if $G(x)=F(ax)$, then we say that $G$ and $F$ belong to the same \emph{scale family} (under $\tau$). \end{itemize}