Basic Concepts
If p follows a beta distribution Bet(α, β), then the odds ratio p/(1-p) follows a beta prime distribution (aka an inverse beta distribution or beta distribution of the second kind).
The pdf of the beta prime distribution is
for x ≥ 0 and α > 0, β > 0, where Beta(α, β) = Γ(α)Γ(β)/Γ(α+β). The cdf is
Solving for x, we obtain, the inverse function, as expected x = p/(1-p). Thus
Key properties
Key statistical properties of the beta distribution are displayed in Figure 1.
Figure 1 – Statistical properties of the beta prime distribution
Worksheet Functions
Real Statistics Functions: Starting with Rel 9.6, the Real Statistics Resource Pack will provide the following worksheet functions:
BETA2_DIST(x, α, β, cum) = the pdf of the beta prime function f(x) when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE.
BETA2_INV(p, α, β) = x such that BETA2.DIST(x, α, β, TRUE) = p; i.e. the inverse of the cdf of the beta prime distribution.
References
Wikipedia (2025) Beta prime distribution
https://en.wikipedia.org/wiki/Beta_prime_distribution
Johnson, N. L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions. Volume 2