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1 Ingeniería y Ciencia ISSN: ingciencia@eafit.edu.co Universidad EAFIT Colombia Zarrazola, Edwin; Nagar, Daya K. Product of independent random variables involving inverted hypergeometric function type I variables Ingeniería y Ciencia, vol. 5, núm. 0, diciembre, 2009, pp Universidad EAFIT Medellín, Colombia Available in: How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative

2 Ingeniería y Ciencia, ISSN Volumen 5, número 0, diciembre de 2009, páginas Product of independent random variables involving inverted hypergeometric function type I variables Produto de variáveis aleatórias independentes envolvendo variáveis com função hipergeométrica invertida tipo I Producto de variables aleatorias independentes que involucran variables con función hipergeométrica invertida de tipo I Edwin Zarrazola and Daya K. Nagar 2 Recepción: 2-jul-2009/Modificación: 08-oct-2009/Aceptación: 09-oct-2009 Se aceptan comentarios y/o discusiones al artículo Abstract The inverted hypergeometric function type I distribution has the probability density function proportional to x ν ( + x (ν+γ 2F (α, β; γ;( + x, x > 0, where 2F is the Gauss hypergeometric function. In this article, we derive the probability density function of the product of two independent random variables having inverted hypergeometric function type I distribution. We also consider several other products involving inverted hypergeometric function type I, beta type I, beta type II, beta type III, Kummer beta and hypergeometric function type I variables. Magíster en matemáticas, edwzar@ciencias.udea.edu.co, profesor asistente, Departamento de Matemáticas, Universidad de Antioquia, Medellín Colombia. 2 PhD in Science, nagar@matematicas.udea.edu.co, profesor titular, Departamento de Matemáticas, Universidad de Antioquia, Medellín Colombia. Universidad EAFIT 93

3 Product of independent random variables involving inverted hypergeometric function type I variables Key words: Appell s first hypergeometric function, beta distribution, Gauss hypergeometric function, Humbert s confluent hypergeometric function, product, transformation. Resumo A distribuição função hipergeométrica invertida tipo I tem a função densidade de probabilidade proporcional a x ν ( + x (ν+γ 2F (α, β; γ;( + x, x > 0, em que 2F é a função hipergeométrica de Gauss. Neste artigo, vamos derivar a função densidade de probabilidade do produto de duas variáveis aleatórias independentes tendo distribuição função hipergeométrica invertida tipo I. Também consideramos vários outros produtos envolvendo variáveis com função hipergeométrica invertida tipo I, beta tipo I, beta tipo II, beta tipo III, Kummer beta e hipergeométrica tipo I. Palavras chaves: primeira funcão hipergeométrica de Appell, distribução beta, função hipergeométrica de Gauss, função hipergeométrica confluente de Humbert, producto, transformação. Resumen La distribución de función hipergeométrica invertida tipo I tiene la función de densidad de probabilidad proporcional a x ν ( + x (ν+γ 2F (α, β; γ;( + x, x > 0, donde 2F es la función hipergeométrica de Gauss. En este artículo se deriva la función de densidad de probabilidad del producto de dos variables aleatorias independientes que se distribuyen según la función hipergeométrica inversa tipo I. También se consideran otros productos entre variables aleatorias con distribución beta tipo I, beta tipo II, beta tipo III, función hipergeométrica tipo I, función hipergeométrica inversa tipo I y Kummer beta. Palabras claves: primera función hipergeométrica de Appell, distribución beta, función hipergeométrica confluente de Humbert, función hipergeométrica de Gauss, producto, transformación. Introduction The random variable X is said to have an inverted hypergeometric function type I distribution, denoted by X IH I (ν,α,β,γ, if its probability density 94 Ingeniería y Ciencia, ISSN

4 Edwin Zarrazola and Daya K. Nagar function(p.d.f. is given by (Gupta and Nagar [], Nagar and Álvarez [2], Γ(γ + ν αγ(γ + ν β x ν ( Γ(γΓ(νΓ(γ + ν α β ( + x ν+γ 2 F α,β;γ;, ( + x where x > 0, ν > 0, γ > 0, γ+ν > α+β, and 2 F is the Gauss hypergeometric function. For α = γ, ( reduces to a beta type II density given by x ν β Γ(γ + ν β, x > 0, Γ(γΓ(ν β ( + x ν β+γ and for β = γ, the inverted hypergeometric function type I density slides to x ν α Γ(γ + ν α, x > 0. Γ(γΓ(ν α ( + x ν α+γ Further, for α = 0 or β = 0, the inverted hypergeometric function type I density simplifies to a beta type II density with parameters ν and γ. Recently, Nagar and Álvarez [2] studied several properties and stochastic representations of the inverted hypergeometric function type I distribution. In this article, we derive the density function of the product of two independent random variables having inverted hypergeometric function type I distribution. We also derive densities of several other products involving hypergeometric function type I, beta type I, beta type II, beta type III, Kummer beta and hypergeometric function type I variables. For further results on product of independent random variables, see: Mathai and Saxena [3], Nagar and Zarrazola [4], and Sanchez and Nagar [5]. In section 2, we give definitions of several univariate distributions. Sections 3 and 4 deal with derivations of a number of densities of products of independent random variables. Finally, in appendix, we give definitions and results on Gauss hypergeometric function, Appell s first hypergeometric function F and Humbert s confluent hypergeometric function Φ. 2 Some Univariate Distributions In this section, we define the beta type I, beta type II, beta type III, hypergeometric function type I and Kummer beta distributions. These definitions Volumen 5, número 0 95

5 Product of independent random variables involving inverted hypergeometric function type I variables can be found in Gordy [6], Johnson, Kotz and Balakrishnan [7], Nagar and Zarrazola [4], and Sánchez and Nagar [5]. Definition 2.. The random variable X is said to have a beta type I distribution with parameters (a,b, a > 0, b > 0, denoted as X B I (a,b, if its p.d.f. is given by {B(a,b} x a ( x b, 0 < x <, where B(a,b is the beta function given by B(a,b = Γ(aΓ(b{Γ(a + b}. Definition 2.2. The random variable X is said to have a beta type II distribution with parameters (a,b, a > 0, b > 0, denoted as X B II (a,b, if its p.d.f. is given by {B(a,b} x a ( + x (a+b, x > 0. Definition 2.3. The random variable X is said to have a beta type III distribution with parameters (a,b, a > 0, b > 0, denoted as X B III (a,b, if its p.d.f. is given by 2 a {B(a,b} x a ( x b ( + x (a+b, 0 < x <. Definition 2.4. The random variable X is said to have a Kummer beta distribution, denoted by X KB(α,β,λ, if its p.d.f. is given by x α ( x β exp [λ( x], 0 < x <, B(α,β F (β;α + β;λ where α > 0, β > 0 and < λ <. Note that for λ = 0 the above density simplifies to a beta type I density with parameters α and β. Definition 2.5. The random variable X is said to have a hypergeometric function type I distribution, denoted by X H I (ν,α,β,γ, if its p.d.f. is given by Γ(γ + ν αγ(γ + ν β Γ(γΓ(νΓ(γ + ν α β xν ( x γ 2F (α,β;γ; x, 0 < x <, where γ + ν α β > 0, γ > 0 and ν > Ingeniería y Ciencia, ISSN

6 Edwin Zarrazola and Daya K. Nagar The following result (Gupta and Nagar [], Nagar and Álvarez [2] states that the hypergeometric function type I distribution can be obtained as the distribution of the product of two independent beta type I variables. Theorem 2.. Let X and X 2 be independent, X i B I (a i,b i, i =,2. Then, X X 2 H I (a,b 2,a + b a 2,b + b 2. The matrix variate generalizations of beta type I, beta type II, beta type III, hypergeometric function type I and Kummer beta distributions have been defined and studied extensively. For example, see Gupta and Nagar [8, ], and Nagar and Gupta [9]. 3 Products of Two Independent Random Variables In this section we derive distributions of the product of two independent random variables when at least one of them has inverted hypergeometric function type I distribution. Theorem 3.. Let X and X 2 be independent random variables, X i IH I (ν i,α i,β i,γ i, i =,2. Then, the p.d.f. of Z = X X 2 is given by { 2 } Γ(ν + γ 2 Γ(ν 2 + γ Γ(γ i + ν i α i Γ(γ i + ν i β i Γ(ν + ν 2 + γ + γ 2 Γ(γ i= i Γ(ν i Γ(γ i + ν i α i β i z ν (α r (α 2 s (β r (β 2 s (ν 2 + γ r (ν + γ 2 s (γ r (γ 2 s (ν + ν 2 + γ + γ 2 r+s r!s! where z > 0. r=0 s=0 2 F (ν + γ 2 + s,ν + γ + r;ν + ν 2 + γ + γ 2 + r + s; z, (2 Proof. Using independence, the joint p.d.f. of X and X 2 is given by K x ν x2 ν2 ( ( ( + x ν+γ ( + x 2 ν2+γ2 2 F α,β ;γ ; 2F α 2,β 2 ;γ 2 ;, + x + x 2 where x > 0, x 2 > 0 and K = 2 i= Γ(γ i + ν i α i Γ(γ i + ν i β i Γ(γ i Γ(ν i Γ(γ i + ν i α i β i. Volumen 5, número 0 97

7 Product of independent random variables involving inverted hypergeometric function type I variables Transforming Z = X X 2, U = /( + X 2 with the Jacobian J(x,x 2 z,u = /u( u in the joint density of X and X 2 and integrating u, we obtain the p.d.f. of Z as K z ν u ν+γ2 ( u ν2+γ ( u 2F α,β ;γ ; 0 [ ( zu] ν+γ ( zu 2 F (α 2,β 2 ;γ 2 ;u du. (3 Now, expanding Gauss hypergeometric functions in the integral (3 in terms of power series we arrive at K z ν (α r (α 2 s (β r (β 2 s u ν+γ2+s ( u ν2+γ+r (γ r (γ 2 s r!s! [ ( zu] ν+γ+r du. r=0 s=0 Finally, using (A.3 and substituting for K we obtain the desired result. Corollary 3... Let X and X 2 be independent random variables, X IH I (ν,α,β,γ, and X 2 B II (ν 2,γ 2. Then, the p.d.f. of Z = X X 2 is given by Γ(ν + γ 2 Γ(ν 2 + γ Γ(γ + ν α Γ(γ + ν β Γ(γ 2 + ν 2 Γ(ν + ν 2 + γ + γ 2 Γ(γ Γ(ν Γ(γ + ν α β Γ(γ 2 Γ(ν 2 z ν (α r (β r (ν 2 + γ r (γ r (ν + ν 2 + γ + γ 2 r r! r=0 0 2 F (ν + γ 2,ν + γ + r;ν + ν 2 + γ + γ 2 + r; z, z > 0. (4 Corollary Let X and X 2 be independent, X i B II (ν i,γ i, i =,2. Then, the p.d.f. of Z = X X 2 is given by Γ(ν + γ 2 Γ(ν 2 + γ Γ(γ + ν Γ(γ 2 + ν 2 Γ(ν + ν 2 + γ + γ 2 Γ(γ Γ(ν Γ(γ 2 Γ(ν 2 z ν 2F (ν + γ 2,ν + γ ;ν + ν 2 + γ + γ 2 ; z, z > 0. (5 Note that the Gauss hypergeometric functions in the densities (2, (4 and (5 can be expanded in series form if 0 < z <. However, if z >, then /z < and we use (A.4 to rewrite the densities (2, (4 and (5 in series involving Gauss hypergeometric functions having /z as argument. The next theorem gives the density of the product of Kummer beta and inverted hypergeometric function type I variables. 98 Ingeniería y Ciencia, ISSN

8 Edwin Zarrazola and Daya K. Nagar Theorem 3.2. Let X and X 2 be independent, X IH I (ν,α,β,γ and X 2 KB(ν 2,γ 2,λ. Then, the p.d.f. of Z = X X 2 is Γ(γ + ν α Γ(γ + ν β Γ(γ 2 + ν 2 Γ(γ + ν 2 Γ(ν Γ(ν 2 Γ(γ Γ(γ + ν α β Γ(γ + γ 2 + ν 2 { F (γ 2 ;ν 2 + γ 2 ;λ} z ν (α r (β r (γ + ν 2 r ( + z ν+γ (γ r=0 + γ 2 + ν 2 r (γ r r!( + z r ] Φ [γ 2,ν + γ + r;γ + γ 2 + ν 2 + r; + z,λ, z > 0. Proof. The joint p.d.f. of X and X 2 is given by x ν x ν2 2 ( x 2 γ2 ( K 2 2F α,β ;γ ; ( + x ν+γ where x > 0, 0 < x 2 < and + x exp[λ( x 2 ], (6 K 2 = Γ(γ + ν α Γ(γ + ν β Γ(γ Γ(ν Γ(γ + ν α β {B(ν 2,γ 2 F (γ 2 ;ν 2 + γ 2 ;λ}. Transforming Z = X X 2, X 2 = X 2 with the Jacobian J(x,x 2 z,x 2 = /x 2 in (6 and integrating x 2 we obtain the marginal p.d.f. of Z as z ν w γ2 ( w γ+ν2 K 2 ( + z ν+γ 0 [ w/( + z] ν+γ ( exp(λw 2 F α,β ;γ ; ( + z ( w dw. (7 w/( + z Now, expanding Gauss hypergeometric functions in the integral (7 in terms of power series we arrive at K 2 z ν (α r (β r ( + z r ( + z ν+γ (γ r r! r=0 0 w γ2 ( w γ+ν2+r exp(λwdw [ w/( + z] ν+γ+r. Finally, applying (A.8 and substituting fork 2 we obtain the desired result. Volumen 5, número 0 99

9 Product of independent random variables involving inverted hypergeometric function type I variables Corollary Let X and X 2 be independent, X B II (ν,γ, and X 2 KB(ν 2,γ 2,λ. Then, the p.d.f. of Z = X X 2 is given by Γ(γ + ν Γ(γ 2 + ν 2 Γ(γ + ν 2 Γ(ν Γ(ν 2 Γ(γ Γ(γ + γ 2 + ν 2 { F (γ 2 ;ν 2 + γ 2 ;λ} [ γ 2,ν + γ ;γ + γ 2 + ν 2 ; zν ( + z ν+γ Φ ] + z,λ, z > 0. Corollary Let X and X 2 be independent, X B II (ν,γ and X 2 B I (ν 2,γ 2. Then, the p.d.f. of Z = X X 2 is given by Γ(γ + ν Γ(γ 2 + ν 2 Γ(γ + ν 2 Γ(ν Γ(ν 2 Γ(γ Γ(γ + γ 2 + ν 2 zν ( + z ν+γ 2 F ( γ 2,ν + γ ;γ + γ 2 + ν 2 ;, z > 0. + z Nagar and Zarrazola [4], while studying the distribution of the product of two independent Kummer beta variables, have also derived the above result. Corollary Let X and X 2 be independent random variables, X IH I (ν,α,β,γ and X 2 B I (ν 2,γ 2. Then, the p.d.f. of Z = X X 2 is given by Γ(γ + ν α Γ(γ + ν β Γ(γ 2 + ν 2 Γ(γ + ν 2 Γ(ν Γ(ν 2 Γ(γ Γ(γ + ν α β Γ(γ + γ 2 + ν 2 zν (α r (β r (γ + ν 2 r ( + z ν+γ (γ r=0 + γ 2 + ν 2 r (γ r r!( + z r 2 F (γ 2,ν + γ + r;γ + γ 2 + ν 2 + r;, z > 0. + z Theorem 3.3. Let the random variables X and X 2 be independent, X IH I (ν,α,β,γ and X 2 B III (ν 2,γ 2. Then, the p.d.f. of Z = X X 2 is given by Γ(γ + ν α Γ(γ + ν β Γ(ν 2 + γ 2 Γ(ν 2 + γ 2 γ2 Γ(ν Γ(ν 2 Γ(γ Γ(γ + ν α β Γ(γ + γ 2 + ν 2 zν (α r (β r (γ + ν 2 r ( + z r ( + z ν+γ (γ r (γ + γ 2 + ν 2 r r! r=0 00 Ingeniería y Ciencia, ISSN

10 Edwin Zarrazola and Daya K. Nagar F (γ 2,ν + γ + r,ν 2 + γ 2 ;γ + γ 2 + ν 2 + r; Proof. The joint p.d.f. of X and X 2 is given by K 3 x ν x2 ν2 ( x 2 γ2 ( + x ν+γ ( + x 2 ν2+γ2 2 F where x > 0, 0 < x 2 < and ( α,β ;γ ; + z,, z > 0. 2 K 3 = Γ(γ + ν α Γ(γ + ν β Γ(γ Γ(ν Γ(γ + ν α β 2ν2 {B(ν 2,γ 2 }., (8 + x Now, transforming Z = X X 2, X 2 = X 2 with the Jacobian J(x,x 2 z,x 2 = /x 2 in (8 and integrating x 2, the marginal p.d.f. of Z is derived as K 3 z ν 2 ν2+γ2 ( + z ν+γ 0 w γ2 ( w ν2+γ [ w/( + z] ν+γ ( w/2 ν2+γ2 ( 2 F α,β ;γ ; ( + z ( w w/( + z dw. (9 Expanding Gauss hypergeometric functions in the integral (9 in series form we arrive at 2 (ν2+γ2 K 3 z ν ( + z ν+γ (α r (β r (γ r r!( + z r r=0 0 w γ2 ( w ν2+γ+r dw [ w/( + z] ν+γ+r ( w/2 ν2+γ2. Finally, the desired result follows by using (A.7 and substituting for K 3. Corollary Let the random variables X and X 2 be independent, X B II (ν,γ and X 2 B III (ν 2,γ 2. Then, the p.d.f. of Z = X X 2 is given by z ν Γ(γ + ν Γ(ν 2 + γ 2 Γ(ν 2 + γ 2 γ2 Γ(ν Γ(ν 2 Γ(γ Γ(γ + γ 2 + ν 2 ( + z γ+ν F (γ 2,ν + γ,ν 2 + γ 2 ;γ + γ 2 + ν 2 ; + z,, z > 0. 2 The above result has also been obtained by Sánchez and Nagar [5]. Volumen 5, número 0 0

11 Product of independent random variables involving inverted hypergeometric function type I variables Theorem 3.4. Let the random variables X and X 2 be independent. Further, X IH I (ν,α,β,γ and X 2 H I (ν 2,α 2,β 2,γ 2. Then, the p.d.f. of Z = X X 2 is given by { 2 } Γ(γ i + ν i α i Γ(γ i + ν i β i B(γ 2,ν 2 + γ Γ(γ i Γ(ν i Γ(γ i + ν i α i β i i= zν ( + z ν+γ s=0 r=0 (α 2 s (β 2 s (α r (β r (ν 2 + γ r ( + z r (γ r (ν 2 + γ + γ 2 s+r s! r! 2 F (γ 2 + s,ν + γ + r;ν 2 + γ 2 + γ + s + r; + z, z > 0. Proof. The joint p.d.f. of X and X 2 is given by K 4 x ν x ν2 2 ( x 2 γ2 ( 2F α,β ;γ ; 2F (α 2,β 2 ;γ 2 ; x 2, ( + x ν+γ + x (0 where 0 < x <, 0 < x 2 < and K 4 = 2 i= Γ(γ i + ν i α i Γ(γ i + ν i β i Γ(γ i Γ(ν i Γ(γ i + ν i α i β i. Now, transforming Z = X X 2, X 2 = X 2 with the Jacobian J(x,x 2 x,z = /x 2 in (0 and integrating x 2, we obtain the p.d.f. of Z as K 4 z ν w γ2 ( w ν2+γ ( ( + z ν+γ 0 [ w/( + z] ν+γ 2 F α,β ;γ ; ( + z ( w w/( + z 2 F (α 2,β 2 ;γ 2 ;w dw, z > 0. Now, expanding the Gauss hypergeometric functions in series form, integrating the resulting expression using (A.3, substituting for K 4 and simplifying, we obtain the desired result. 4 Products of Three Independent Random Variables In this section we derive distributions of products of three independent random variables involving inverted hypergeometric function type I, beta type I and beta type II variables. 02 Ingeniería y Ciencia, ISSN

12 Edwin Zarrazola and Daya K. Nagar Theorem 4.. Let X,X 2 and X 3 be independent random variables, X i B I (a i,b i, i =,2 and X 3 IH I (ν,α,β,γ. Then, the p.d.f. of Z = X X 2 X 3 is given by Γ(a + b Γ(a 2 + b 2 Γ(a + γ Γ(γ + ν αγ(γ + ν β Γ(a Γ(a 2 Γ(a + b + b 2 + γ Γ(νΓ(γΓ(γ + ν α β zν ( + z ν+γ r=0 s=0 (b 2 s (a + b a 2 s (α r (β r (a + γ r ( + z r (γ r (a + b + b 2 + γ s+r s!r! 2 F (b + b 2 + s,ν + γ + r;a + b + b 2 + γ + s + r; + z, z > 0. Proof. Using Theorem 2., X X 2 H I (a,b 2,a + b a 2,b + b 2. Now, using independence of X,X 2 and X 3 and Theorem 3.4, we obtain the desired result. Corollary 4... Let X,X 2, and X 3 be independent, X i B I (a i,b i, i =,2 and X 3 B II (a 3,b 3. Then, the p.d.f. of Z = X X 2 X 3 is given by Γ(a + b Γ(a 2 + b 2 Γ(a 3 + b 3 Γ(a + b 3 Γ(a Γ(a 2 Γ(a 3 Γ(a + b + b 2 + b 3 Γ(b 3 za3 (b 2 s (a + b a 2 s ( + z a3+b3 (a + b + b 2 + b 3 s s! s=0 2 F (b + b 2 + s,a 3 + b 3 ;a + b + b 2 + b 3 + s;, z > 0. + z Appendix: Additional Definitions and Results Here we give some definitions and additional results which are used throughout this work. We use the Pochammer symbol (a n defined by (a n = a(a + (a + n = (a n (a + n for n =,2,..., and (a 0 =. The generalized hypergeometric function of scalar argument is defined by pf q (a,...,a p ;b,...,b q ;z = k=0 (a k (a p k z k (b k (b q k k!, (A. Volumen 5, número 0 03

13 Product of independent random variables involving inverted hypergeometric function type I variables where a i, i =,...,p; b j, j =,...,q are complex numbers with suitable restrictions and z is a complex variable. Conditions for the convergence of the series in (A. are available in the literature, see Luke [0]. From (A. it is easy to see that and 0F 0 (z = k=0 z k k! = exp(z, F (a;c;z = 2F (a,b;c;z = k=0 (a k (b k (c k k=0 z k, z <. k! (a k z k (c k k!, The integral representations of the confluent hypergeometric function and the Gauss hypergeometric function are given as and F (a;c;z = 2F (a,b;c;z = Γ(c Γ(aΓ(c a Γ(c Γ(aΓ(c a 0 0 t a ( t c a exp(ztdt, t a ( t c a ( zt b dt, arg( z < π, (A.2 (A.3 respectively, where Re(a > 0 and Re(c a > 0. It is easy to check by using (A.3 that 2F (a,b;c;z = ( z a 2F (a,c b;c; The Appell s first hypergeometric function F is defined by F (a,b,b 2 ;c;z,z 2 = = = r,s=0 r=0 s=0 (a r+s (b r (b 2 s z rzs 2 (c r+s r!s! z. (A.4 z (a r (b r (c r z r r! 2 F (a + r,b 2 ;c + r;z 2 (a s (b 2 s (c s z s 2 s! 2 F (a + s,b ;c + s;z, (A.5 04 Ingeniería y Ciencia, ISSN

14 Edwin Zarrazola and Daya K. Nagar where z < and z 2 <. The Humbert s confluent hypergeometric function Φ is defined by Φ [a,b ;c;z,z 2 ] = = = r,s=0 r=0 s=0 (a r+s (b r (c r+s z r zs 2 r!s!, (a r (b r (c r z r r! F (a + r;c + r;z 2 (a s (c s z s 2 s! 2 F (a + s,b ;c + s;z, (A.6 where z <, z 2 <. The integral representations of F and Φ are given by Γ(c v a ( v c a dv F (a,b,b 2 ;c;z,z 2 =, (A.7 Γ(aΓ(c a 0 ( vz b b2 ( vz 2 where z <, z 2 <, Re(a > 0 and Re(c a > 0, and Φ [a,b ;c;z,z 2 ] = Γ(c Γ(aΓ(c a 0 v a ( v c a exp(vz 2 dv ( vz b, (A.8 where z <, Re(a > 0 and Re(c a > 0. Note that for b = 0, F and Φ reduce to 2 F and F functions, respectively. For properties and further results on these functions the reader is referred to Luke [0], Srivastava and Karlsson [], Mathai and Saxena [2], and Mathai [3]. Acknowledgments This research work was supported by the Comité para el Desarrollo de la Investigación, Universidad de Antioquia research grant no. IN550CE. References [] A. K. Gupta and D. K. Nagar. Matrix variate distributions, ISBN Chapman & Hall/CRC. Boca Raton, Referenced in 95, 97 Volumen 5, número 0 05

15 Product of independent random variables involving inverted hypergeometric function type I variables [2] Daya K. Nagar and José A. Álvarez. Properties of the hypergeometric function type I distribution. Advances and Applications in Statistics, ISSN , 5(3, (2005. Referenced in 95, 97 [3] A. M. Mathai and R. K. Saxena. Distribution of product and the structural set up of densities. Annals of Mathematical Statistics, ISSN , 40(4, (969. Referenced in 95 [4] D. K. Nagar and E. Zarrazola. Distributions of the product and the quotient of independent Kummer beta variables. Scientiae Mathematicae Japonicae, ISSN , eissn , 6(, 09 7 (2005. Referenced in 95, 96, 00 [5] Luz E. Sánchez and Daya K. Nagar. Distributions of the product and quotient of independent beta type 3 variables. Far East Journal of Theoretical Statistics, ISSN , 7(2, (2005. Referenced in 95, 96, 0 [6] Michael B. Gordy. Computationally convenient distributional assumptions for common-value auctions. Computational Economics, ISSN , eissn , 2(, 6 78 (998. Referenced in 96 [7] N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions-2, second edition, ISBN John Wiley & Sons, New York, [8] A. K. Gupta and D. K. Nagar. Matrix variate beta distribution. International Journal of Mathematics and Mathematical Sciences, ISSN 06 72, eissn , 24(7, (2000. Referenced in 97 [9] D. K. Nagar and A. K. Gupta. Matrix variate Kummer beta distribution. Journal of the Australian Mathematical Society, ISSN , eissn , 73(, 25 (2002. Referenced in 97 [0] Y. L. Luke. The special functions and their approximations, vol., ISBN X. Academic Press, New York, 969. Referenced in 04, 05 [] H. M. Srivastava and P. W. Karlsson. Multiple Gaussian hypergeometric series, ISBN Halsted Press [John Wiley & Sons], New York, 985. Referenced in 05 [2] A. M. Mathai and R. K. Saxena. The H-function with applications in statistics and other disciplines, ISBN X. Halsted Press [John Wiley & Sons], New York, 978. Referenced in 05 [3] A. M. Mathai. A handbook of generalized special functions for statistical and physical sciences, ISBN Oxford University Press, New York, 993. Referenced in Ingeniería y Ciencia, ISSN