BETA PRIME DISTRIBUTION

Author

LAKSHMINARASIMHAN P S

ABSTRACT

This paper in general defines one of the major probability distribution function used in probability theory and statistics. A distribution in statistics is a function that shows the possible values for a variable and how often they occur. Each distribution is characterized by its shape, central tendency and variability. The distribution of an event consists not only of the input values that can be observed, but is made up of all possible values. Each probability distribution is associated with a graph describing the likelihood of occurrence of every event. Usually, these scores are arranged in order from smallest to largest and then they can be presented graphically.

INTRODUCTION

Beta prime distribution also known as inverted beta distribution, beta distribution of second kind, beta type II, compound gamma, gamma ratio, inverse (or inverted) beta, Pearson type VI is an absolutely continuous probability distribution. It is the distribution of the odds ratio associated with a random variable with the beta distribution. It is defined on the interval [0, ∞). The distribution has fat tails which decrease polynomially. In Bayesian statistics the distribution is a conjugate family of prior distributions on the odds parameter of the binomial distribution.

Beta Prime Distribution[p,q,α,β] represents a continuous statistical distribution defined over the interval and parametrized by four positive real numbers p, q, α, and β. The parameters p, q, and α are known as “shape parameters”, β is known as a “scale parameter”, and together, these parameters determine the overall shape of the probability density function (PDF) of the beta prime distribution. Depending on the values of p, q, α, and β, the PDF of the beta prime distribution may be uni-modal or monotonic decreasing with potential singularities approaching the lower boundary of its domain. In addition, the tails of the PDF are “fat” in the sense that the PDF decreases algebraically rather than exponentially for large values x.

Beta Prime Distribution[p,q,α,β] is sometimes referred to as the generalized beta distribution of the second kind, the inverted beta distribution, or the type VI Pearson distribution. The two- and three-argument forms Beta Prime Distribution[p,q] and Beta Prime Distribution[p,q,β] evaluate to Beta Prime Distribution[p,q,1,1] and Beta Prime Distribution[p,q,1,β], respectively, and are sometimes referred to as the standard beta prime distribution and the generalized beta prime distribution, respectively.

In Bayesian analysis, the beta prime distribution arises as a prior distribution for binomial proportions expressed as odds. The beta prime distribution has also been found to model many real-world phenomena. For example, the beta prime distribution has proven useful in empirically estimating security returns and in the development of option pricing models. More recently, it has been applied to the modeling of insurance loss processes. Elsewhere, the long tail of the beta prime distribution has been shown to make the distribution particularly well suited to modeling the frequency of behaviors likely to transmit diseases among individuals versus the actual transmission of such diseases.

BETA PRIME DISTRIBUTION

DEFINITION

Suppose that U has the beta distribution with shape parameters a,b ∈(0,∞). Random variable X=U/(1-U) has the beta prime distribution with shape parameters a and b. Beta prime distribution is defined for with two parameters α and β, having the probability density function:

where B is the Beta function.

The cumulative distribution function is

where I is the regularized incomplete beta function.

The mean is α(β – 1) for β > 1.
The mode is (α – 1)/ (α + β – 2) for α > 1, β > 1.
The median cannot be expressed in a simple closed form expression.

Its non-central moments (for integral ) are:

Generalization:

Two more parameters can be added to form the generalized beta prime distribution

  • p > 0 shape (real)
  • q > 0 scale (real)

having the probability density function:

with mean:

and mode:

Note: If p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution.

FORMULAS

DISTRIBUTION CURVES

Probability density function

Cumulative distribution function

APPLICATIONS

It is used in many applications, that includes

  • Bayesian hypothesis testing

  • The rule of succession

  • Task duration modelling

  • Project planning control systems like CPM and PERT

  • Beta Prime Distribution can be used to model state per-capita incomes

  • Beta Prime Distribution can be used to model losses

PROBLEM

Question:

Suppose that the lifetime of a certain kind of an emergency backup battery (in hours) is a random variable X having the Beta Prime Distribution with α = 1 and β = 5 find

  1. The mean lifetime of these batteries

  2. The mode lifetime of these batteries when α = 0

  3. The probability such that a battery will last more than 3 hours

Solution:

CONCLUSION

We can conclude that the Beta Prime Distribution is one of the powerful distributions that can be used to find continuous probability distribution either as in its original form or as its special case distributions such as power function distribution ( where B is negative ) or as Burr distribution, Dagum distribution, F-distribution, Log-logistic distribution and Lomax distribution.

REFERENCE

  • https://en.wikipedia.org/wiki/Beta_prime_distribution

  • https://www.statisticshowto.com/beta-prime-distribution-2/

  • http://www.nematrian.com/BetaPrimeDistribution

  • http://www.scientificlib.com/en/Mathematics/LX/BetaPrimeDistribution.html