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Hometype of a distribution function
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type of a distribution function
Two distribution functions $F,G:\mathbb{R}\to[0,1]$ are said to of the same type if there exist $a,b\in\mathbb{R}$ such that $G(x)=F(ax+b)$. $a$ is called the scale parameter, and $b$ the location parameter or centering parameter. Let’s write $F\stackrel{t}{=}G$ to denote that $F$ and $G$ are of the same type.
Remarks.

Necessarily $a>0$, for otherwise at least one of $G(\infty)=0$ or $G(\infty)=1$ would be violated.

If $G(x)=F(ax)$, then the graph of $G$ is stretched from the graph of $F$ by $a$ units if $a>1$, and compressed if $a<1$.

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
$P(X\leq z)=F(z)=G(\frac{zb}{a})=P(Y\leq\frac{zb}{a})=P(aY+b\leq z).$ When $X$ and $aY+b$ are identically distributed, we write $X\stackrel{t}{=}Y$.

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, $Var[X]=a^{2}Var[Y]$. In general, convergence of moments is a “typical” property.

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.

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 $\tau$.

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 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.

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 location family under $\tau$; and if $G(x)=F(ax)$, then we say that $G$ and $F$ belong to the same scale family (under $\tau$).
Mathematics Subject Classification
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