The Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model the distribution of incomes and other financial variables.
In the following discussion, will denote the Pareto distribution as defined above. As will be shown below, the exponential distribution is considered a light tailed distribution. Yet mixing exponentials produces the heavy tailed Pareto distribution. Mixture distributions tend to heavy tailed (see ). The Pareto distribution is a handy example.
Note that the shape parameter of the Pareto distribution, , equals , where is the power law slope. Also note that for there is no finite mean for the distribution. Presumably because of this, the Pareto distribution is sometimes given with , but the definition is more widely used.
Open the and select the Pareto distribution. Vary the shape parameter and note the shape of the probability density function. For selected values of the parameter, run the simulation 1000 times and compare the empirical density function to the probability density function.