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'type of a distribution function'
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| Title of object: |
type of a distribution function |
| Canonical Name: |
TypeOfADistributionFunction |
| Type: |
Definition |
| Created on: |
2006-11-22 22:50:34 |
| Modified on: |
2006-11-24 22:40:39 |
| Classification: |
msc:60E05, msc:62E10 |
| Defines: |
type, scale factor, location factor, standard distribution function, location family, scale family |
| Synonyms: |
type of a distribution function=centering factor
type of a distribution function=scale parameter
type of a distribution function=location parameter |
Revision comment (for changes between this and next version):
| Changes for correction #10983 ('clarification'). |
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Content:
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 For any equivalence class of distribution functions belonging to a certain type $\tau$, there is one distribution function $F$ that corresponds 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} |
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