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[parent] type of a distribution function (Definition)

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
    $\displaystyle P(X\le z)=F(z)=G(\frac{z-b}{a})=P(Y \le \frac{z-b}{a})=P(aY+b \le z).$
    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$).



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Other names:  centering factor, scale parameter, location parameter
Also defines:  type, scale factor, location factor, standard distribution function, location family, scale family

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Cross-references: normal distribution, variance, equivalence class, probability space, fixed, real, equivalence relation, binary relation, functions, subsets, disjoint, partition, property, moments, finite, expectation, iff, easy to see, identically distributed, random variables, units, right, graph, parameter, distribution functions
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This is version 10 of type of a distribution function, born on 2006-11-22, modified 2006-11-29.
Object id is 8582, canonical name is TypeOfADistributionFunction.
Accessed 4361 times total.

Classification:
AMS MSC: 60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
 62E10 (Statistics :: Distribution theory :: Characterization and structure theory)
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