Beta prime distribution

June 02, 2026 Aleksejus Kononovicius #interactive #statistics #voter model #power-law distributions #Kirman model #R. Kazakevicius

Have you heard of the beta prime distribution before? Until recently, I hadn't either. My colleague, Rytis Kazakevičius, recently surprised me by pointing out that the distribution we have been repeatedly encountering in nonlinear transformations of the noisy voter model has actually a proper name. It is known as the beta prime distribution. And our history with this distribution, goes back much further.

Context and derivation

It goes back to the agent-based herding model of financial markets [1]. In that paper we have used Kirman model to simulate the switching behavior between chartist and fundamentalist trading strategies. While the fraction of chartists (or fundamentalists) is distributed according to the Beta distribution (as Kirman model is a variation of the voter model), the absolute returns generated by the model did follow some kind of power-law distribution. That distribution is exactly the beta prime distribution.

In the agent-based herding model of financial markets [1] we have shown that under the Walrasian market assumption the long-term log-returns of the financial markets can be modeled as a ratio of chartists and fundamentalists. If we let \( x \) to denote the fraction chartists in the market (then the fraction of fundamentalists is given by \( 1-x \)), then the long-term log-returns will be given by

\begin{equation} y = \frac{x}{1-x} . \label{eq:main-transform} \end{equation}

This transformation not only allows us to explore the dynamics in the financial markets, but also heterogeneous diffusion process in general. In a series of more recent papers [2, 3, 4] we have used it as a basis for a generalized nonlinear transformation of the noisy voter model. Though notably, the beta prime distribution arises only for the transformation given by \eqref{eq:main-transform}.

Now, if the fraction of agents (lets say chartists) \( x \) is distributed according to \( \mathcal{B}e \left(\alpha,\beta\right) \), then from the conservation of probability mass,

\begin{equation} p_Y(y) d y = p_X(x) d x , \end{equation}

and for our particular case, we have that,

\begin{equation} p_Y(y) = p_X\left(\frac{y}{1+y}\right) \left| \frac{d}{dy} \left(\frac{y}{1+y}\right) \right| = \frac{y^{\alpha-1}\left(1+y\right)^{-\alpha-\beta}}{B\left(\alpha,\beta\right)} . \end{equation}

Which is exactly the probability density function of the beta prime distribution.

Interactive app

Use the interactive app below to explore how the beta-distributed random values get transformed into the beta prime-distributed values. The left plot of the app shows the probability density function of the generated beta random variables, while the right plot shows the probability density function of the transformed variables. In both plots, the gray curves represent the corresponding theoretical probability density functions.

References

A. Kononovicius, V. Gontis. Agent based reasoning for the non-linear stochastic models of long-range memory. Physica A 391: 1309-1314 (2012). doi: 10.1016/j.physa.2011.08.061. arXiv: 1106.2685 [q-fin.ST].
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].