Beta prime distribution from Gamma-distributed random values

As we have already discussed, beta prime distribution arises from a nonlinear transformation of the voter model. Furthermore, recently we have been relying a lot on the said transformation [1, 2, 3]. In those papers, we have been using some results derived for the CIR process as well. Thus, another interesting thing, which my colleague Rytis Kazakevičius has noted, was that the beta prime distribution can be obtained from the ratio of two independent Gamma-distributed random values. Why it is interesting? The stationary distribution of the CIR process is the Gamma distribution!

I will not delve into analytical derivation of this result this time, just let me share the interactive app with you. The left plot of the app shows the probability density functions of the two generated Gamma-distributed random variables. The right plot shows the probability density function of their ratio. In both plots, the gray curves represent the corresponding theoretical probability density functions (Gamma distribution on the left, and beta prime distribution on the right).

Note that the beta prime distribution is obtained only if \( \theta_1 = \theta_2 \) (for any positive values). If the shape parameter values are different, the probability density function of the ratio will deviate from the beta prime distribution. Though, if the shape parameter values are different, the ratio is still distributed according to a generalized beta prime distribution.

References

• R. Kazakevicius, A. Kononovicius. Anomalous diffusion in nonlinear transformations of the noisy voter model. Physical Review E 103: 032154 (2021). doi: 10.1103/PhysRevE.103.032154. arXiv:2011.02927 [cond-mat.stat-mech].
• R. Kazakevičius, A. Kononovicius. Anomalous diffusion and long-range memory in the scaled voter model. Physical Review E 107: 024106 (2023). doi: 10.1103/PhysRevE.107.024106. arXiv:2301.08088 [cond-mat.stat-mech].
• R. Kazakevicius, A. Kononovicius. Mean first passage time of the symmetric noisy voter model with arbitrary initial and boundary conditions. Chaos, Solitons and Fractals 203: 117649 (2026). doi: 10.1016/j.chaos.2025.117649. arXiv:2512.02519 [cond-mat.stat-mech].