Normal Approximating Whatever

I finally got back to a math book, Modeling the Dynamics of Life by Frederick R. Adler,  I’ve had it on hold for a long while. I’ve been at it for over a year. And, I still haven’t done the homework. The homework actually teaches beyond the text in a lot of math books. So I’ll be at it for a long time to come even though I’m starting the final chapter. It’s an applied textbook, so the author gets his point across without turning you into a mathematician, or at least tries to. The mathematician thing will happen if you pay attention, but who does that?

In the previous chapter, the book talks about approximating a Poisson distribution with a normal. That’s a very small normal since it fits inside that Poisson distribution it’s trying to approximate. It does the same sort of thing for the Binomial. And, again for the exponential. I drew the series of distributions for this latter exercise. It takes a lot of distributions added together to get that normal, a lot like 30 distributions. The thing that can get lost is the shape of the world holding the distribution.

In approximating the normal from an exponential, the exponential, aka long tail looked longer than it was tall. But adding two distributions brought us to a gamma distribution that was a little longer. Adding five distributions got us something that looked normal, but was wider still, and pdf was taller than the normal. Adding ten distributions, wider again and less tall. Adding 30, wider, practically on top of each other and shorter. If we kept on adding, it would get shorter and wider, aka it would get tiny, but the approximation and the actual would be close enough that we’d be collecting data and graphing things for entertainment.

This graph will be too small. But take a look.

Sum of Distributions Tending to Normal

At some point further calculation becomes pointless. Factor analysis shares this property. Does another factor tell you something actionable? Does more accuracy do the same?

Another thing that got talked about was the standard normal. You get to the standard normal from the normal via z-scores. You want all your distributions to have a normal approximation since your tools for approximating probabilities are based on the standard normal and its z-scores. To do hypothesis testing, you need a normal.

You can find those formulas for distributions. They tend to look messy. Try integrating them. Getting to a standard normal is easier. Another author in another book that I can’t cite, said that while the numbers convert via those formulas, the logic does not follow the flow of the calculations. Hypothesis testing in non-normal distributions is an active area of research. An example of calculation and logic not being consistent,  we have  machine learning, Markovian approaches discover, while Gaussian approaches enforce. That’s not really a matter of application. One is ontological while the other approach is taxonomic.

Notice that all these approximations and converging tos require a lot of data and a lot of distributions. We are using big data to estimate small data.

Enjoy! Comments?


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