Pareto distribution of the second kind | Vose Software

# Pareto distribution of the second kind

Format: Pareto2(b, q)

This distribution is simply a standard Pareto distribution (of the first kind) but shifted along the x-axis so that it starts at x = 0. This is most readily apparent by studying the cumulative distribution functions for the two distributions:

 Pareto: Pareto2:

The only difference between the two equations is that x for the Pareto has been replaced by (x+b) for the Pareto2. In other words:

VosePareto2(b,q) = VosePareto(q,a) - a

where a = b, and q = q

Thus both distributions have the same variance and shape when a = b and q = q, but different means.

## Uses

A Pareto2 (1/b, a) distribution describes the time between events where the number of random events occurring in a unit of time follows a Pólya (a,b) distribution.

This is more than an academic curiosity. It is commonly assumed that events occurring randomly in time follow a Poisson distribution, from which it can be determined that the time between events follows an Exponential distribution. However, many events seem to show some clustering in their timing, which means that there are more events closer together and further apart than the Exponential distribution would predict. Fitting the time between events using both a Pareto2 and an Exponential distribution, then comparing which distribution fits better, allows one to assess whether a Pólya or a Poisson should be used for modeling frequency.

One might argue - why not just fit the frequency data in the first place? The problem with frequency data is that it is information-poor. If we know when events occurred, we can summarize how many occurred in each time interval, but the reverse is not possible. Thus, by fitting distributions to time between events we are making the most of the available data and are more readily able to determine whether a Poisson or Pólya process is more appropriate.

## ModelRisk functions added to Microsoft Excel for the Pareto distribution of the Second Kind

VosePareto2 generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VosePareto2Object constructs a distribution object for this distribution.

VosePareto2Prob returns the probability density or cumulative distribution function for this distribution.

VosePareto2Prob10 returns the log10 of the probability density or cumulative distribution function.

VosePareto2Fit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VosePareto2FitObject constructs a distribution object of this distribution fitted to data.

VosePareto2FitP returns the parameters of this distribution fitted to data.