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 $G(x)=F(x+b)$ then the graph of $G$ is shifted to the right from the graph of $F$ by $b$ units, if $b>0$ and to the left if $b<0$
• 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 $$P(X\le z)=F(z)=G(\frac{z-b}{a})=P(Y \le \frac{z-b}{a})=P(aY+b \le z).$$ When $X$ and $aY+b$ are identically distributed, we write $X \stackrel{t}{=} Y$
• 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.
• 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 $\tau$ .