Normal distributions are nice and easy but alas, they do not always explain reality. Sometimes, this lack of "normality" can lead to major problems when models are built on them. This path is not a complete reference, but can get you started in at least seeing it is a problem.
Fat-tail and long-tail probability analysis is used in financial risk management, but it would seem that statisticians and investment professionals would sometimes disagree on what constitues a fat tail and what is a long tail. See the two diagrams below.
Edit Remove MoveA look at statistical roots of crisis
Edit Remove MoveThis is really good. Highly recommended. A look in: "A tail event is an outcome which, from the perspective of the frequency of historical events or perhaps only from intuition, should happen only once in a thousand or million or centillion years. Tail events are more than statistical curiosities. In some cases, they may be so important that they dominate the way we think about our options and our strategies"
Edit Remove MoveA fat-tailed distribution is a probability distribution that has the property, along with the other heavy-tailed distributions, that it exhibits large skewness or kurtosis. This comparison is often made relative to the normal distribution, or to the exponential distribution. Fat-tailed distributions have been empirically encountered in a variety of areas: economics, physics, and earth sciences.
Edit Remove MoveIn a normal distribution, the largest number of observations centers around the mean and declines to the right and left as there are typically fewer observations of extreme returns (for instance -30%) thus giving it a bell shape. Units of standard deviation (‘Sigma’ noted by the symbol δ in the diagram) provide a measurement of the expected frequency of the returns. In a normal distribution, returns that are generated within one sigma of the mean are predicted to happen 68.5% of the time. Fat tails exist because extreme events are far more prevalent than is explained by the normal distribution."
Edit Remove MoveA normal distribution varies a lot in the neighborhood of its average, but produces few examples beyond three standard deviations from that average. Normal distributions are common in biology. For example, men average 5' 10," and their population has a standard deviation of 4".
Edit Remove MoveTaleb is the one person more associated with Fat Tails than anyone I know. So I will give him the final word: Background: The technical papers below are part of a systematic approach to uncover mismeasurement of statistical estimators under fattailedness and propose corrections and alternative tools. Conventional statistics fail to cover fat tails; physicists who use power laws do not usually produce statistical estimators, leading to a large -and consequential - gap.
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