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Suppose one has a conventionally parameterized Normal Distribution in which the first parameter is a and the second parameter is b. Such a distribution can be expressed as a transform of the base Normal Distribution in which the first parameter is 0 and the second parameter is 1. In particular, the transform is $a + b * x$. My conjecture is that some distributions have this useful property -- i.e. any member of the family can be expressed as some algebraic transformation of any other member -- and some distributions do not.

a) Is my conjecture correct or do all distributions have this property? b) Does the property I am describing have some sort of name?

Response to Question

In response to the question first asked in response to this post, let me see if I can clarify

$a + b x$ means you transform the output of the underlying distribution by adding a to it and multiplying by b. Yes, you can translate and scale any PDF, but I don't think that is what I am asking about.

Maybe another example would help. Suppose we have a distribution such as the Pareto Distribution, conventionally parameterized, in which the first parameter is, say, 3 and the second parameter is 4. Call this distribution g. Just for clarify, the PDF of g is $\begin{cases} \frac{324}{x^5} & x\geq 3 \\ 0 & \text{True} \end{cases}$. There's obviously another Pareto Distribution f with a first parameter of 1 and a second parameter of 1. Again, for clarity, the PDF of f is $\begin{cases} \frac{1}{x^2} & x\geq 1 \\ 0 & \text{True} \end{cases}$.

There is a transformation of f that is the exact equivalent of g? It is $3 \sqrt[4]{x}$. That is, if one takes an output x from f and applies the function $3 \sqrt[4]{x}$ the resulting random variable is the same that is obtained from g. (I did this, by the way, by having Mathematica evaluate the following expression: Quantile[ParetoDistribution[3, 4], CDF[ParetoDistribution[1, 1], x]]). But again, is this always possible. I am beginning to think that it might be. And what is the property or term that one uses to describe this property?

My guess is that it might be "homo" something. Or "closed under transformations" but I really don't know. Hope this helps.

mathlawguy
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  • What do you mean by $a+bx$? What are you transforming? In general, you can translate and scale any pdf (although one would need to also scale in the vertical direction to guarantee that the area under the curve is $1$). – Michael Burr Aug 16 '15 at 17:58
  • Most families of distributions will not give you another member of the same family after a scale and a shift (you may need more parameters to model this situation). I am not aware of a name for this property because it describes more about the types of models we've given names to than the properties of the models themselves. – Michael Burr Aug 16 '15 at 21:18

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