< List of probability distributions < *Beta prime distribution*

The **Beta Prime Distribution** is a continuous probability distribution defined on the interval [0, ∞). The distribution has polynomially decreasing fat tails; this tells us that that the probability of observing an extreme value decreases as a polynomial function of the distance from the mean.

In Bayesian statistics, the distribution is a conjugate prior distribution on the odds parameter of the binomial distribution. The *odds parameter* is the ratio of the probability of success to failure in a binomial experiment.

The beta prime distribution has many other names, including: *beta type II*, *compound gamma*, *gamma ratio*, *inverse (or inverted) beta*, *Pearson type VI*, and *variance ratio*. The beta prime is a special case of the type 6 Pearson distribution, but it is not the only special case. Other special cases of the type 6 Pearson distribution, which are also Pearson type VI, include the gamma distribution, inverse gamma distribution and the F distribution. The beta prime and F distribution are closely related by a scale transformation.

## Beta prime distribution properties

The standardized probability density function (PDF) for the beta prime distribution is:

With shape parameters α, β, and Beta function *Β*(α, β). Some definitions do include a scale parameter, λ, but this is not common; Most definitions are defined for λ = 1 [2].

- The cumulative distribution function (CDF) is: F(x) = Ix/(1+x)(α, β).
- 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.

The relationship between the beta distribution and the beta prime is as follows: If a random variable *Z* is from a beta distribution, then *X* =* Z* – 1 – 1 is from the beta prime distribution [2]. The relationship can be defined as

**P(X = x) = betaprime(x; α + 1, β + 1)**

where **betaprime(x; α + 1, β + 1) **is the PDF of the beta prime with α + 1 and beta + 1.

## Connection with the F distribution

If a random variable *X* follows an *F* distribution with *n* ∈ (0, ∞) numerator degrees of freedom and *d* ∈ (0, ∞) denominator degrees of freedom, then *Y* = (*n*/*d*) *X* has a beta prime distribution with parameters *n*/2 and *d*/2. Similarly, if *Y* is beta prime with parameters *a* ∈ (0, ∞) and *b* ∈ (0, ∞), then *X* = (b/a)X is *F*-distributed with 2*a *numerator degrees of freedom and 2*b* denominator degrees of freedom [3].

## Beta prime distribution special cases

The beta prime distribution has many special cases, overlapping with some families of probability distributions. For example, the power function distribution is a special case of the beta prime distribution when *β *is negative. Other notable special cases include [4]:

## References

[1] Fishman, G. (2001). Discrete-Event Simulation: Modeling, Programming, and Analysis. Springer.

[2] Laurent, S. (2019). R-Bloggers–The Beta Distribution of the Third Kind. Retrieved December 31, 2021 from: https://www.r-bloggers.com/2019/07/the-beta-distribution-of-the-third-kind-or-generalised-beta-prime/

[3] Random Services. The F-Distribution. Retrieved July 20, 2023 from: https://www.randomservices.org/random/special/Fisher.html

[4] Crooks, G. Survey of Simple, Continuous, Univariate Probability Distributions. Retrieved December 31, 2021 from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.372.3694&rep=rep1&type=pdf