The Bell Curve. Murray should have used the indefinite article "A" instead of the definite article "The." This is but one of many examples that suggest that most scientists simply equate the entire infinite set of probability bell curves with the normal bell curve of textbooks.
Actually, in a specific context, for example, standardized test scores taken from a population of students at random, there is only one distribution of scores; and it is quite likely that this is what Murray implied. Moreover, the generalization - from Murray's book- to most scientists is breathtakingly absurd...
Human and non-human behavior can be far more diverse than the classical normal bell curve allows.
Not if the conditions of the Central Limit Theorem are met, and they are met under a (not really) surprisingly broad array of conditions.
Again most bell curves have thick tails.
If guys like this can get published...anyway, most bell curves...? Look, he just got through (correctly) saying in effect that the number of bell curves has the same cardinality as the contiuum - that there are as many bell curves as there are points on the real line. It follows therefore that there are "as many" bell curves - namely C- as there are points on the real line, that have "thin" tails (whatever that means) as have "thick" tails.
The first reason it fails is that the classical central limit theorem result rests on a critical assumption that need not hold and that often does not hold in practice...Most bell curves have infinite or undefined variance even though they have a finite dispersion about their center point.
The major assumption of the Central Limit Theorem is boundedness of moments; in effect finite energy. Ignoring the repeated stupidity about "most bell curves," Kosko makes a crucial error here: probability is about observation, and the our ability to observe, happens to have the limitation of finite dynamic range - that is, our measurement meters only go up to V volts, for example.
Stable or thick-tailed probability curves continue to turn up as more scientists and engineers search for them. They tend to accurately model impulsive phenomena such as noise in telephone lines or in the atmosphere or in fluctuating economic assets.
They are be non-Gaussian, but "stable" does not equal "thick-tailed," for reasons evident from above...
...Whiteness just means that the noise spikes or hisses and pops are independent in time or that they do not correlate with one another....
Actually it means something far more profound than that, that the sample functions of the white noise are everywhere continuous but nowhere differentiable...in otherwords, it is impossible to visualize a sample function of white noise except as an abstract mathematical object.
The real "dangerous ideas" are:
1. Why are so many things in nature represented by stable distributions (that is the distribution of the sum of random variables representing quantities from nature are the same class of random variables as the individual quantity)? The short answer is "because we can represent them that way."
2. Why are so many things in nature actually indescribable by probability distributions?
I gotta get back into publishing research if guys like Kosko can make up stuff like this...