A continuous variable with a long tail distribution | Vose Software

# A continuous variable with a long tail distribution

What do we mean by a long-tailed distribution? One distribution is said to have a longer tail than another if its probability density (or mass) function is (asymptotically) larger than the other distribution's for very large values of the variable, i.e. for two distributions A and B:

Many socioeconomic and other natural random variables take long-tailed distributions. Examples are city population sizes, occurrences of natural resources (e.g. size of reserves in a certain geological region), stock price fluctuations, size of companies, income.

The most commonly fitted distribution to the extreme of such data has been the Pareto. There is no decent theory to explain why the Pareto distribution tends to fit the tails of long-tailed variables, but most people accept that it works and use it anyway.

The Pareto is usually a poor fit for the main body of the variable, though. Thus, when modelling long-tailed distributions one usually does so using a splice of one distribution (like the Lognormal, or Gamma, for example), with a Pareto distribution to model the tail.

In ModelRisk you can use the Splicing Distributions window to splice two distributions together.