The other bell curves

After lines, conic sections (circles, parabolas, etc.), and graphs of trigonometric functions, the most familiar curve has to be the bell curve.  Perhaps it is even more famous than some or all of those other curves, especially amongst students who just finished an exam that they don't feel good about.

Explaining exactly why we use the bell curve takes some work, but a rough explanation can be given in terms of coin flips.  If you flip a coin say 20 times, then you can expect to get 20 heads with a certain probability, 19 heads and 1 tail with a certain probability, 18 heads and 2 tails with a certain probability, and so on.  This is an example of a binomial distribution.  If you plot these percentages, you get a shape that looks roughly like the bell curve.  For 30 coin flips, the shape looks even more like the bell curve.

In fact, you can make the shape looks as close to the bell curve as you want by considering a large enough number of coin flips.  The coin doesn't even need to be fair.  That is, it could favour one of heads or tails more than the other, and with enough coin flips, you could still make the shape look as much like the bell curve as you like.

So lurking behind many things that give a shape that looks like the bell curve, there is a process something like a coin flip going on.  This is an oversimplification, of course.  There are other ways to reach the bell curve besides coin flips.  The whole story would take a while to explain, however.  The point is that we can arrive at the bell curve through a simple, familiar process.

The most famous bell curve is not the only bell curve, though.  The one I'm talking about above is known as the Normal Curve or Gaussian Curve [1], but there are other curves besides these that have a bell shape, but aren't Normal Curves.

One curve comes from the logistic distribution.  Another curve comes from the Cauchy distribution.  What I wonder about these curves is this.  Is there is some way to arrive at either of these distributions from some simple process in the same way that we can arrive at the normal distribution through the process of coin flips?

One place where the logistic distribution crops up is in the ELO rating system.  This system was developed by a physicist named Arpad Elo (for some reason, his name gets capitalized, as if it's an acronym, when it comes to the rating system) to rate chess players, though it can be used for any type of competition, from individual games like chess to team sports, like soccer.  Each competitor has a rating, and the probability that one competitor will beat another can be calculated from their ratings.  The calculation depends a probability distribution.  In the original scheme, a normal distribution was used, but it was discovered that a logistic distribution gave more accurate predictions.

In fact, it was this application of the logistic distribution that led me to my question above.  The empirical evidence suggests that it's the better choice, but I want to know if there is a reason why.  Is there some sort of logistic equivalent to coin flipping for the normal distribution that explains why predictions are better with the logistic distribution?


Logistic curves also pop up in some population growth models.  There are continuous and discrete versions of these models, which in a sense parallel the normal curve (continuous) and the coin flips (discrete).  Perhaps the answer to my question lies here, but I can't see the connection between population growth and games.  (Both do involve competition, I suppose, but it's not clear to me whether or not this is a meaningful connection, or just a linguistic coincidence.)

The Wikipedia article for the Cauchy distribution says that "it has the distribution of a random variable that is the ratio of two independent standard normal random variables."  That suggests that ratios of two independent binomial random variables (coin flips, for example) could be an avenue to explore.

It's also not hard to create other bell curves.  Take any function $f(x)$ that is positive, symmetric about some vertical axis, and concave up.  Then $1/f(x)$ will give a bell curve.  I expect that most of these aren't interesting, though.  Are there any others, besides the 3 mentioned above, that are?

[1] It's actually a class of curves that are related to each other by stretching and shifting vertically and horizontally.