Distribution Name: Power Law
Parameters:
- Alpha (Shape parameter)
- Theta (Location parameter)
The power law is a useful distribution for describing phenomena with extreme, catastrophic possibilities—even more than lognormal. For events such as forest fires, the vast majority of occurrences are limited to an acre or less in scope. On rare occasions, however, a forest fire may spread over hundreds of acres. The “fat tail” of the power law distribution allows us to acknowledge the common small event, while still accounting for more extreme possibilities.
- When to Use: When you want to make sure that catastrophic events, while rare, will be given nontrivial probabilities.
- Examples: Phenomena like earthquakes, power outages, epidemics, and other types of “cascade failures” have this property.
- Excel Formula: =(theta/x)^alpha
- Mean: =(alpha*theta/(alpha-1))