# Tools for self-study: Constructing and understanding systems of distributional families

I am looking for a resource that probably does not exist, but, well, hope springs eternal.

I have become increasingly interested in the process by which distributions are discovered or invented. Examples would include:

1. The systems of Pearson and Burr, rooted in differential equations;
2. Compound distributions;
3. Systems of distributions built up from simpler distributions by various transformations;
4. Sums and products of distributions;
5. Ways of looking at candidate functions and defining the parameter restrictions required for them to exist, and useful tricks for deriving the necessary constants to set the integral to one;
6. Ways of finding distributions that have a certain property, such as one of the various forms of self-similarity;
7. Distribution that are the limit of repeated applications of some specified form of stochastic shock.

Now of course, there is a huge literature on all these things, so why am I saying I doubt it exists? I think these are, usually but not always, regarded as advanced topics, at least outside of a statistics major. Most of them, for instance, are either not covered or are relegated to a problem or two in Hogg & Craig. I am looking for a treatment of these topics, suitable for self-study, at perhaps the advanced undergraduate level.

Take the example of the Pearson and Burr systems. I know that each of these families of distributions are derived as the set of solutions to a differential equation. But I don't understand what motivates the choice of those particular differential equations, that is, I don't have the intuitions on which they are based; and I could certainly not go from the differential equations to finding all the distributions derived from it, that is, I lack the technical tools to do concrete derivations. I was pretty good at calculus once -- I still think Apostol is the best textbook I have ever encountered, in any discipline -- but that was a long time ago. Today, I would have to see more intermediate results and worked-out examples in order to follow a text on this topic. A solutions manual would be helpful. I want a text that does not assume, e.g., that I already know how to do a convolution or find a non-trivial limiting distribution.

But I am pretty sure I do not want a graduate statistics text. For instance, I am willing to assume without proof that the sets I encounter will be measurable, and that Riemann integration will be adequate to my needs.

Actually, as I described what I hoped for, I persuaded myself that there could be such a text or other resource, or even several such texts, aimed at junior or senior undergraduate statistics majors. Hope springs eternal.

• so what is it that you want? I came up with a "new" distribution once while calculating the probabilities of certain events. i had to formulate the problem, then solved it, incorrectly, and got an equation that described some probabilities that added up to 1. I called it a distribution. why not? – Aksakal Jun 18 '18 at 17:28