Pearson Type 5 distribution | Vose Software

# Pearson Type 5 distribution

Format: Pearson5(a, b)

The  Pearson family of distributions was designed by Karl Pearson between 1890 and 1895. It represents a system whereby for every member the probability density function f(x) satisfies a differential equation:

(1)

where the shape of the distribution is depends on the values of the parameters a, c0, c1, and c2. The Pearson Type V corresponds to the case where c0 + c1x + c2x2 is a perfect square (c2=4c0c2). Thus, equation (1) can be rewritten as:

Examples of the Pearson Type V distribution are given below:

## Uses

The Pearson Type 5 distribution has been usefully used to model time delays where there is almost certainty of some minimum delay and the maximum delay is unbounded, for example: delay in arrival of emergency services and time to repair some machine (see Law and Kelton 1991, p339. Cummins et al (1990) have used the Pearson Type 5 distribution to model fire losses at a university.

## Notes

The Pearson Type 5 distribution is also often called the Inverse Gamma distribution (and sometimes the Inverted Gamma distribution (mostly in reliability modeling) or the Reciprocal Gamma distribution) because it has a reciprocal relationship to the Gamma distribution as follows:

Pearson5(a,1/b) = 1/Gamma(a,b)

The 'Inverted Gamma' distribution is very popular in Bayesian statistics as an uninformed prior for modeling variance of a random variable.

A Pearson5 distribution with a = 0.5 is a Lévy distribution, of particular interest because it is a stable distribution.

The MaxEnt uncertainty distribution for a parameter with known mean and harmonic mean is a Pearson5.

The Pearson family includes many familiar distributions:

• The Normal distribution

• Beta distribution, Inverse Beta distribution (=1/Beta), Gamma distribution, and Inverse Gamma distribution (=1/Gamma) which usually have an overall bell-shape but are generally skewed left or right

• Student t distributions, which are symmetrical (unskewed) but have longer tails than the Normal distribution

• Type II distributions, which are symmetric but have thicker, shorter tails than the Normal distribution. The Uniform distribution is of Type II

## ModelRisk functions added to Microsoft Excel for the Pearson Type 5 distribution

VosePearson5 generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VosePearson5Object constructs a distribution object for this distribution.

VosePearson5Prob returns the probability density or cumulative distribution function for this distribution.

VosePearson5Prob10 returns the log10 of the probability density or cumulative distribution function.

VosePearson5Fit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VosePearson5FitObject constructs a distribution object of this distribution fitted to data.

VosePearson5FitP returns the parameters of this distribution fitted to data.