It’s amazing how easy it is to study mathematical statistics in great detail without learning a thing about which distributions fit which natural (or unnatural) phenomena. I found this piece on Distribution Fitting, from StatSoft‘s online textbook remarkably helpful:
Variables whose values are determined by an infinite number of independent random events will be distributed following the normal distribution, whereas variables whose values are the result of an extremely rare event would follow the Poisson distribution. The major distributions that have been proposed for modeling survival or failure times are the exponential (and linear exponential) distribution, the Weibull distribution of extreme events, and the Gompertz distribution. The section on types of distributions contains a number of distributions generally giving a brief example of what type of data would most commonly follow a specific distribution as well as the probability density functin (pdf) for each distribution.
I must admit, I hadn’t even heard of some of these distributions. The Weibull distribution sounds fascinating:
The Weibull distribution is often used in the field of life data analysis due to its flexibility — it can mimic the behavior of other statistical distributions such as the normal and the exponential. If the failure rate decreases over time, then k < 1. If the failure rate is constant over time, then k = 1. If the failure rate increases over time, then k > 1.An understanding of the failure rate may provide insight as to what is causing the failures:
- A decreasing failure rate would suggest “infant mortality”. That is, defective items fail early and the failure rate decreases over time as they fall out of the population.
- A constant failure rate suggests that items are failing from random events.
- An increasing failure rate suggests “wear out” — parts are more likely to fail as time goes on.
When k = 3.4, then the Weibull distribution appears similar to the normal distribution. When k = 1, then the Weibull distribution reduces to the exponential distribution.
Hmm… I suspect I really geeked out there — even more than normal.