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Gauss’ Ghost

Carl GaussJohann Carl Friedrich Gauss was a polymath of no mean skill. Mathematicians bemoan the fact that he spent his later years doing physics, and physicists wish he had started earlier. One of his contributions was the derivation and proofs for the bell-shaped curve known as a “Gaussian” or “normal” distribution. It is the result of a random process in which small steps are taken in any direction. So universal is the “Gaussian” in all areas of life that it is taken to be prima facie evidence of a random process.

Only in recent years have people addressed situations that can deviate from a Gaussian. For example, one of the criteria that produce a Gaussian, is that the probability of a “small” step must be greater than the probability of a “big” step. That is, if we consider the random walk of the proverbial drunk near a lightpole, if he staggers in small steps most of the time, a plot of his position taken, say, every other second, would be a Gaussian. But if he staggers in big steps with a few small ones thrown in, then the plot begins to look peculiar. Instead of being a Gaussian, it develops a “fat tail“, with many locations far from the lightpole.

Now why is this important? Because many people predict that Darwinian evolution is driven by random processes of small steps. This implies that there must be some Gaussians there if we knew where to look.

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