# On the product of the bivariate beta components

- 156 Downloads

## 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.

## Keywords

Product of random variables Bivariate beta distribution Beta distribution## 1 Introduction

For given random variables \(X_{1}\) and \(X_{2}\), the distribution of the product \(P:=X_{1}X_{2},\) arise in many fields as biology, economics, engineering, genetics, hydrology, medicine, number theory, order statistics, physics, etc. (see, for example, Frisch and Sornette 1997; Sornette 1998; Nadarajah 2008). It is one of the most important research areas both from theoretical and application points of view.

In a number of applications, it is necessary to specify the properties of the product of random variables: this occurs in particular when the dimension of the involved random variables are of ratio type as for fuel consumption per mile, cost of a structure per 1 lb. of payload, amplification ratio, tolerances expressed in percentages of the desired value, etc. For instance, if the number of accidents during a period can be regarded as a random variable and if the same applies to the number of days spent in hospital by an accident victim and to the total cost per one day-patient, then the total cost is equal to the product of these three random variables. Another example is the following: if \(X_{i}\) is the random variable describing the amplification of the \(i\hbox {th}\) amplifier, then the total amplification \(X_{1},X_{2},\ldots ,X_{n}\) is also a random variable and it is important to know the distribution of this product.

Important examples in economics have be considered in detail by Nadarajah (2008). In fact, Nadarajah remarked that in traditional portfolio selection models, the product of random variables is often involved. The best examples are investments in different foreign markets. In portfolio diversification models (see, for example, Grubel 1968), there is uncertainty in prices of shares in local markets and uncertainty about exchange rates make the value of the portfolio in domestic currency related to a product of random variables. Similarly, in diversified production models by multinational companies (see, for example, Rugman 1979), not only is the local production is uncertain but also the exchange rates are uncertain so that the profits in local currency are again related to a product of random variables. Furthermore, another example drawn from the econometric literature is the following: in providing a forecast from an estimated equation, Feldstein (1971) pointed out that both the parameter and the value of the exogenous variable during the forecast period could be considered as random variables. The forecast was therefore proportional to a product of random variables.

In physics, Frisch and Sornette (1997) have developed a theory of extreme deviations generalizing the central limit theorem which, when applied to multiplication of random variables, predicts the generic presence of stretched exponential probability distribution functions (pdf’s). Their problem comes down to determine the tail of the pdf for a product of random variables.

Product of random variables also arising in hydrology stream flow is often defined as a product of two or more variables, representing, for example, the periodic and the stochastic components, respectively (see Cigizoglu and Bayazit 2000).

The distribution of the product \(P=X_{1}X_{2}\) has been studied by many researchers and for many distributions, among them Sakamoto (1943) for the uniform distribution, Springer and Thompson (1970) for the normal distribution, Tang and Gupta (1984) for the beta distribution and Nadarajah and Kotz (2004) for the Dirichlet distribution.

*P*and show its robustness.

*A*has the beta distribution with parameters \(a,b>0\) (say \(A \overset{d}{\sim }\)\(\beta (a,b)),\) then its density function is

## 2 The bivariate beta distribution

## 3 Product of the bivariate beta components

The aim of this work is to study the product \(P:=X_{1}X_{2}\) when \(X_{1}\) and \(X_{2}\) are distributed according to Eq. (1). We derive in this section the exact density and moment functions of *P* and we give an approximation of its distribution. Here, we show goodness of its robustness.

### 3.1 Density function of *P*

In the following result, we derive the exact density function of the product *P*.

### Theorem 1

*If*\(X_{1}\)

*and*\(X_{2}\)

*are jointly distributed according to*(1),

*then the density function of*\(P=X_{1}X_{2},\)

*is given by*

*for*\(0<p<1.\)

### Proof

*P*can be written as

### 3.2 Moments

In this section, we give the moments of the product \(P:=X_{1}X_{2}\). Now we establish the following result.

### Theorem 2

*If*\(X_{1}\)

*and*\(X_{2}\)

*are jointly distributed according to*(1),

*then the moment function of*\(P=X_{1}X_{2},\)

*is given by*

*for*\(n\ge 1\).

*In particular, the first two moments of*

*P*

*are*

*and*

### 3.3 Approximation

*P*has support in the interval \(\left[ 0,1\right] .\) From this, it is evident that we will motivate to approximate its distribution by the beta distribution with parameters \(a,b>0\) (say \(P\overset{d}{\sim }\)\(\beta (a,b))\) and density function

*a*and

*b*is made using the method of moments. The first two moments of

*P*can be written as

Estimates of (*a*, *b*) for selected \(\left( \theta _{1},\theta _{2},\theta _{3}\right)\)

\(\theta _{1}\) | \(\theta _{2}\) | \(\theta _{3}\) | | |
---|---|---|---|---|

\(\begin{array}{c} 0.5 \\ 0.5 \\ 2 \\ 4 \\ 4 \\ 4 \\ 4 \\ 6 \end{array}\) | \(\begin{array}{c} 0.5 \\ 2 \\ 2 \\ 2 \\ 4 \\ 2 \\ 4 \\ 6 \end{array}\) | \(\begin{array}{c} 2 \\ 2 \\ 2 \\ 2 \\ 2 \\ 4 \\ 4 \\ 6 \end{array}\) | \(\begin{array}{c} 0.1718 \\ 0.3567 \\ 0.9889 \\ 1.3575 \\ 2.0834 \\ 1.2703 \\ 1.9729 \\ 2.9642 \end{array}\) | \(\begin{array}{c} 3.2541 \\ 2.7357 \\ 2.6127 \\ 2.4623 \\ 2.4008 \\ 5.8306 \\ 5.5056 \\ 8.4547 \end{array}\) |

*a*and

*b*using (10) and (11). From this, we use the selected parameters \(\left( \theta _{1},\theta _{2},\theta _{3}\right)\) and the estimates are shown in Table 1. Next, we checked robustness by comparing the exact and the approximated density functions of

*P*as given by (3) and (9). These comparisons are illustrated in Figs. 1 and 2. It is clear that the approximation is quite good, and we remark that for (3) in Fig. 1, the solid curve is unimodal, and the broken curve is monotone decreasing. Similar findings were noted when this exercise was repeated for many other combinations of \(\left( \theta _{1},\theta _{2},\theta _{3}\right) .\) Noting that the numerical results are obtained by the use of PYTHON and the hyp2f1 and hyp3f2 functions in mpmath 0.14.

## Notes

### 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.

## References

- A-Grivas, D., & Asaoka, A. (1982). Slope safety prediction under static and seismic loads.
*Journal of Geotechnical and Geoenvironmental Engineering*,*108*, 713–729.Google Scholar - Apostolakis, F. J., & Moieni, P. (1987). The foundations of models of dependence in probabilistic safety assessment.
*Reliability Engineering*,*18*, 177–195.CrossRefGoogle Scholar - Chatfield, C. (1975). A marketing application of a characterization theorem.
*Statistical Distributions in Scientific Work*,*2*, 175–185.CrossRefGoogle Scholar - Cigizoglu, H. K., & Bayazit, M. (2000). A generalized seasonal model for flow duration curve.
*Hydrological Processes*,*14*, 1053–1067.CrossRefGoogle Scholar - Das Gupta, P. (1968). Two approximations for the distribution of double non-central beta.
*Sankhyā*,*30*, 83–88.MathSciNetGoogle Scholar - Fan, D.-Y. (1991). The distribution of the product of independent beta variables.
*Communications in Statistics-Theory and Methods*,*20*, 4043–4052.MathSciNetCrossRefGoogle Scholar - Feldstein, M. S. (1971). The error of forecast in econometric models when the forecast-period exogenous variables are stochastic.
*Econometrica*,*39*, 55–60.CrossRefzbMATHGoogle Scholar - Frisch, U., & Sornette, D. (1997). Extreme deviations and applications.
*Journal de Physique I France*,*7*, 1155–1171.CrossRefGoogle Scholar - Grubel, H. G. (1968). Internationally diversified portfolios: Welfare gains capital flows.
*The American Economic Review*,*58*, 1299–1314.Google Scholar - Gupta, A. K., & Nadarajah, S. (2006). Exact and approximate distributions for the linear combination of inverted Dirichlet components.
*Journal of the Japan Statistical Society*,*36*, 225–236.MathSciNetCrossRefzbMATHGoogle Scholar - Hoyer, R. W., & Mayer, L. S., (1976). The equivalence of various objective functions in a stochastic model of electoral competition. Department of Statistics, Princeton University. Tech. Rep. 114, Series 2.Google Scholar
- Johannesson, B., & Giri, N. (1995). On approximations involving the beta distribution.
*Communications in Statistics-Simulation and Computation*,*24*, 489–503.MathSciNetCrossRefzbMATHGoogle Scholar - Libby, D. L., & Novick, M. R. (1982). Multivariate generalized beta-distributions with applications to utility assessment.
*Journal of Educational Statistics*,*7*, 271–294.CrossRefGoogle Scholar - Nadarajah, S. (2006). The bivariate \(F_{3}\)- beta distribution.
*Communications of the Korean Mathematical Society*,*21*, 363–374.CrossRefzbMATHGoogle Scholar - Nadarajah, S. (2007). A new bivariate beta distribution with application to drought data.
*Metron*,*2*, 153–174.MathSciNetGoogle Scholar - Nadarajah, S. (2008). On the product of generalized Pareto random variables.
*Applied Economics Letters*,*15*, 253–259.CrossRefGoogle Scholar - Nadarajah, S., & Kotz, S. (2004). Exact and approximate distributions for the product of Dirichlet components.
*Kybernetika*,*40*(6), 735–744.MathSciNetzbMATHGoogle Scholar - Olkin, I., & Liu, R. (2003). A bivariate beta distribution.
*Statistics and Probability Letters*,*62*, 407–412.MathSciNetCrossRefzbMATHGoogle Scholar - Prudnikov, A. P., Brychkov, Y. A., & Marichev, O. I. (1986).
*Integrals and series (volumes 1 and 3)*. Amsterdam: Gordon and Breach Science Publishers.zbMATHGoogle Scholar - Rugman, A. M. (1979).
*International diversification and the multinational enterprise*. Lexington, Mass: Lexington Books.Google Scholar - Sakamoto, H. (1943). On the distributions of the product and the quotient of the independent and uniformly distributed random variables.
*Tohoku Mathematical Journal*,*49*, 243–260.MathSciNetzbMATHGoogle Scholar - Sculli, D., & Wong, K. L. (1985). The maximum and sum of two beta variables in the analysis of PERT networks.
*Omega*,*13*, 233–240.CrossRefGoogle Scholar - Sornette, D. (1998). Multiplicative processes and power laws.
*Physical Review E*,*57*, 4811–4813.CrossRefGoogle Scholar - Springer, M. D., & Thompson, W. E. (1970). The distribution of products of beta, gamma and Gaussian random variables.
*SIAM Journal on Applied Mathematics*,*18*, 721–737.MathSciNetCrossRefzbMATHGoogle Scholar - Tang, J., & Gupta, A. K. (1984). On the distribution of the product of independent beta random variables.
*Statistics and Probability Letters*,*2*, 165–168.MathSciNetCrossRefzbMATHGoogle Scholar