On the product of the bivariate beta components

Abstract

The aim of the present paper is to derive the exact distribution and the corresponding moment function of the product \(P:=X_{1}X_{2}\) when \(X_{1}\) and \(X_{2}\) are distributed according to a bivariate beta distribution. We also give approximation for this distribution and show its robustness.

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Acknowledgements

The author would like to thank the coordinating editor and the referee for carefully reading the paper and for their comments which greatly improved the paper.

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Correspondence to M. Ghorbel.

Appendix 1

Appendix 1

Using the series expansion

$$\begin{aligned} \left( 1-z\right) ^{-\alpha }=\sum _{k=0}^{\infty }\frac{\Gamma \left( \alpha +k\right) z^{k}}{\Gamma \left( \alpha \right) k!}, \end{aligned}$$

one can write

$$\begin{aligned} \mathbf {E}\left( X_{1}^{q_{1}}X_{2}^{q_{2}}\right)= & {} \frac{\Gamma \left( \sum _{m=1}^{3}\theta _{m}\right) }{\prod _{m=1}^{3}\Gamma \left( \theta _{m}\right) }\int _{0}^{1}x_{1}^{q_{1}+\theta _{1}-1}\left( 1-x_{1}\right) ^{\theta _{2}+\theta _{3}-1}\int _{0}^{1}\frac{x_{2}^{q_{2}+\theta _{2}-1}\left( 1-x_{2}\right) ^{\theta _{1}+\theta _{3}-1}}{\left( 1-x_{1}x_{2}\right) ^{\sum _{m=1}^{3}\theta _{m}}}dx_{2}dx_{1}\nonumber \\= & {} \frac{\Gamma \left( \sum _{m=1}^{3}\theta _{m}\right) }{ \prod _{m=1}^{3}\Gamma \left( \theta _{m}\right) }\int _{0}^{1}x_{1}^{q_{1}+ \theta _{1}-1}\left( 1-x_{1}\right) ^{\theta _{2}+\theta _{3}-1}B(\theta _{2}+q_{2},\theta _{1}+\theta _{3}) \nonumber \\&\times\, _{2}F_{1}(\theta _{2}+q_{2},\sum _{m=1}^{3}\theta _{m},q_{2}+\sum _{m=1}^{3}\theta _{m},x_{1})dx_{1}. \end{aligned}$$
(12)

The result of Eq. (2) follows using Eq. (2.21.1.4) in Prudnikov et al. (1986, volume 3) to calculate the integral in (12).

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Ghorbel, M. On the product of the bivariate beta components. Jpn J Stat Data Sci 2, 71–80 (2019). https://doi.org/10.1007/s42081-018-0027-1

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Keywords

  • Product of random variables
  • Bivariate beta distribution
  • Beta distribution