< 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 2a numerator degrees of freedom and 2b 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