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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 11:34:11 |
| 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 |
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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}.
\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 $$Y=\frac{X-b}{a},$$ since $P(Y\le z)=G(z)=F(az+b)=P(X\le az+b)=P(\frac{X-b}{a}\le z)$.
\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 ``of the same type'' 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, 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}; and if $G(x)=F(ax)$, then we say that $G$ and $F$ belong to the same \emph{scale family}.
\end{itemize} |
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