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Format: VosePearson5(a, b, U)
The Pearson family of distributions was designed by 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:
The Pearson5 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 Pearson5 (aka Inverse Gamma) distribution to model fire losses at a university.
The Pearson5 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 Levy 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, Inverse Beta (=1/Beta), Gamma, and Inverse Gamma (=1/Gamma) distributions 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
VosePearson5 generates values from this distribution or calculates a percentile
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
