3.9. Rare Events#

This section deals with random variables that can be modelled with probability distributions with tail events that are not that rare, and possibly with extreme outcomes, like natural disasters or terrorist events1. These probability distributions have heavy or fat tails, not decaying as fast as tails of a normal distribution do. As a consequence of their very nature, these variables may show some weird behavior:

  • moments may not exists, or may be infinite: some of these distributions have infinite variance, some have not-defined expected value,…

  • results as the central limit theorem don’t hold, as the required assumptions are not satisfied by these distributions

  • …

Author’s Note

The motivation for writing this section comes from a suggestion by a friend, who pointed me to a Veritasium video. It was something I had been meaning to do for a long time, like many other things, for that matter. However, watching that video provided a concrete opportunity to finally sit down and start writing this section.

It is likely that this section will also draw inspiration from the podcast La logica del rischio by P. Cirillo, at least as far as the examples are concerned. Since this is a work in progress, more specific references will be added as the work develops.

Contents

Introduction to heavy-tailed distributions.

Power-law distributions.

References
1

[Estimating the probability of rare events] (https://aaronclauset.github.io/rareevents/), A.~Clauset adn R.~Woodard.