Question

I'm interested in collecting real-world examples of probability distributions, and heuristics for when a probability distribution might apply.

For example:

1. Stock prices are often modeled with lognormal distributions, assuming stock returns are normally distributed. More generally, lognormal distributions often apply when there's a proportional effect going on; that is, when the change in object X is proportional to object X's current size.

2. If your data is skewed (e.g., prices are skewed to the right, since they're required to be positive), then a lognormal distribution might be good to try.

Answer 1

T. Tao wrote a very interesting article on Zipf/Pareto distributions.

Answer 2

Of course the normal (Gaussian) distribution comes up frequently due to the central limit theorem. However, the normal distribution usually only describes the "middle" well; its tails are often too thin to model the probability of rare events well.

Stable distributions, such as Cauchy, come up when aggregating power law data due to a generalized central limit theorem.

Survival time with constant hazard: exponential.

Survival time with increasing or decreasing hazard: Weibull.

Angela Duckworth argues that "achievement" (nice, broad term) is often distributed as lognormal in a nice 90-second video.

Count data is often Poisson distributed, unless there's overdispersion and then negative binomial may fit better.