Beta prime distribution

< List of probability distributions

The Beta Prime Distribution is a continuous probability distribution defined on the interval [0, ∞). The distribution has polynomially decreasing fat tails. In Bayesian statistics, the distribution is a conjugate family of prior distributions on the odds parameter of the binomial distribution.

PDF of the beta prime distribution.

The distribution has many other names, including: beta type II, compound gamma, gamma ratio, inverse (or inverted) beta, Pearson type VI, and variance ratio. The relationship between the beta distribution and beta prime is as follows: If a random variable Z is from a beta distribution, the X = Z-1 – 1 is from the beta prime distribution [2].

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

PDF of the beta prime distribution.

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 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.

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 [3]:

  • Burr distribution,
  • Dagum distribution,
  • Log-logistic distribution,
  • Lomax distribution,
  • F-distribution.


[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:

[3] Crooks, G. Survey of Simple, Continuous, Univariate Probability Distributions. Retrieved December 31, 2021 from: