Bivariate Generalization of The Inverted Hypergeometric Function Type I Distribution

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1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No. 3,, ISSN Bivariate Generalization of The Inverted Hypergeometric Function Type I Distribution Paula A. Bran-Cardona, Edwin Zarrazola and Daya K. Nagar, Departamento de Matemáticas, Universidad del Valle, Calle 3, No. -, Cali, Colombia Instituto de Matemáticas, Universidad de Antioquia, Calle 67, No. 53-8, Medellín, Colombia Abstract. The bivariate inverted hypergeometric function type I distribution is defined by the probability density function proportional to x ν x ν + x + x (ν +ν +γ) F (α,β;γ;(+ x + x ) ), x >, x >, whereν,ν,α,β andγare suitable constants. In this article, we study several properties of this distribution and derive density functions of X /X, X /(X + X ) and X + X. We also consider several products involving bivariate inverted hypergeometric function type I, beta type I, beta type II, beta type III, Kummer-beta and hypergeometric function type I variables. Mathematics Subject Classifications: 6E5, 6E5 Key Words and Phrases: Appell s first hypergeometric function, Beta distribution, Gauss hypergeometric function, Humbert s confluent hypergeometric function, product, transformation.. 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 function (p.d.f.) is given by Nagar and Alvarez[8], Γ(γ+ν α)γ(γ+ν β) x ν Γ(γ)Γ(ν)Γ(γ+ν α β) (+ x) ν+γ F α,β;γ;, x>, () + x whereν>,γ>,γ+ν>α+β, and F is the Gauss hypergeometric function. Forα=γ, the density () reduces to a beta type II density given by Corresponding author. Γ(γ+ν β) Γ(γ)Γ(ν β) x ν β, x>, (+ x) ν β+γ addresses: ÔÙкÖÒÑкÓÑ (Bran-Cardona), ÞÖÖÞÓÑкÓÑ (Zarrazola), ÝÒÖÝÓÓºÓÑ (Nagar) 37 c EJPAM All rights reserved.

2 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), and forβ=γ, the inverted hypergeometric function type I density slides to Γ(γ+ν α), x>. Γ(γ)Γ(ν α) (+ x) ν α+γ x ν α Further, forα=orβ=, the inverted hypergeometric function type I density simplifies to a beta type II density with parametersν andγ. Recently, Nagar and Alvarez[8] studied several properties and stochastic representations of the inverted hypergeometric function type I distribution. Zarrazola and Nagar[6] derived the density function of the product of two independent random variables having inverted hypergeometric function type I distribution. They 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. The bivariate generalization of the inverted hypergeometric function type I distribution, denoted by(x, X ) IH I (ν,ν ;α,β,γ), is defined by the density[nagar, Bran-Cardona and Gupta 9], x ν x ν C(ν,ν ;α,β,γ) ν +ν + x + x +γ F α,β;γ;, () + x + x where x >, x >, and C(ν,ν ;α,β,γ) is the normalizing constant given by C(ν,ν ;α,β,γ)= Γ(ν +ν +γ α)γ(ν +ν +γ β) Γ(ν )Γ(ν )Γ(γ)Γ(ν +ν +γ α β), withν >,ν >,γ>, andν +ν +γ>α+β. Forα=orβ=, the density () slides to a Dirichlet type II density of order 3 with parametersν,ν andγ. It can also be observed that bivariate generalization of the hypergeometric function type I distribution defined by the density () belongs to the Liouville family of distributions proposed by Marshall and Olkin[6] and Sivazlian[4]. In this article we study several properties of the bivariate generalization of the hypergeometric function type I distribution defined by the density (). In Section, we derive results such as the marginal and the conditional densities, moments and correlation and in Section 3 we show that if(x, X ) IH I (ν,ν ;α,β,γ), then, X + X H I (ν +ν,α,β,γ), which is independent of X /(X + X ) B I (ν,ν ) and X /X B I I (ν,ν ). In Section 4, we derive density functions of(x X 3, X X 3 ), where(x, X ) and X 3 are independent,(x, X ) IH I (ν,ν ;α,β,γ) and (i) X 3 IH I (κ,µ,ρ,σ), (ii) X 3 B I I (κ,σ), (iii) X 3 KB(κ,µ,λ) (iv) X 3 B I (κ,µ),

3 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), (v) X 3 B I I I (κ,µ), and (vi) X 3 H I (κ,µ,ρ,σ). Finally, in appendix we give definitions and results on Gauss hypergeometric function, Appell s first hypergeometric function F, Humbert s confluent hypergeometric functionφ and statistical distributions.. Properties In this section we study several properties of the bivariate distribution defined in Section. We first derive marginal and conditional distributions. Theorem. If(X, X ) IH I (ν,ν ;α,β,γ), then the p.d.f. of X is given by Γ(ν +γ)γ(ν +ν +γ α)γ(ν +ν +γ β) Γ(ν )Γ(γ)Γ(ν +ν +γ)γ(ν +ν +γ α β) x ν (+ x ) ν +γ 3F α,β,ν +γ;γ,ν +ν +γ;, + x x >. (3) Proof. By integrating x in (), we get the marginal p.d.f. of X as x ν C(ν,ν ;α,β,γ) (+ x ) ν +γ z ν+γ ( z) ν z F α,β;γ; dz, + x where we have used the substitution z=(+ x )/(+ x + x ). Now, the desired result is obtained by using (A.). Forα=γ, the density () reduces to x ν x ν Γ(ν +ν )Γ(ν +ν +γ β) Γ(ν )Γ(ν )Γ(γ)Γ(ν +ν β) β ν +ν x + x + x + x +γ β, (4) where x > and x >. The marginal density of X in this case is given by Γ(ν +γ)γ(ν +ν )Γ(ν +ν +γ β) Γ(ν )Γ(γ)Γ(ν +ν +γ)γ(ν +ν β) x ν (+ x ) ν +γ F β,ν +γ;ν +ν +γ;, + x x >. (5) Using the above theorem, the conditional density function of X given X = x > is obtained as Γ(γ+ν +ν ) x ν (+ x ) γ+ν F (α,β;γ;(+ x + x ) ) Γ(ν )Γ(γ+ν )(+ x + x ) γ+ν +ν 3F (α,β,γ+ν ;γ,γ+ν +ν ;(+ x ) ),

4 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), where x > and x >. Further, using (), the joint(r,s)-th moment is obtained as E(X r X s )= C(ν,ν ;α,β,γ) x ν +r x ν +s (+ x + x ) ν +ν +γ F α,β;γ; + x + x Now, substituting u= x /(x + x ), v=x + x and z= /(+ v) with the Jacobian J(x, x u,z)=j(x, x u, v)j(v z)=( z)/z 3 in the above integral, one obtains E(X r X s )= C(ν,ν ;α,β,γ)b(ν + r,ν + s) dx dx. z γ r s ( z) ν +ν +r+s F (α,β;γ; z) dz. Finally, evaluating the above integral using (A.) and simplifying the resulting expression, we get E(X r X s ) = Γ(ν +r)γ(ν + s)γ(γ r s)γ(ν +ν +γ α)γ(ν +ν +γ β) Γ(ν )Γ(ν )Γ(γ)Γ(ν +ν +γ)γ(ν +ν +γ α β) 3 F (α,β,γ r s;γ,ν +ν +γ; ), whereν + r>,ν + s> andγ> r+ s. Now, substituting appropriately, we obtain E(X i ) = ν i Γ(ν +ν +γ α)γ(ν +ν +γ β) γ Γ(ν +ν +γ)γ(ν +ν +γ α β) 3 F (α,β,γ ;γ,ν +ν +γ; ), E(X i ) = ν i (ν i + ) Γ(ν +ν +γ α)γ(ν +ν +γ β) (γ )(γ ) Γ(ν +ν +γ)γ(ν +ν +γ α β) 3 F (α,β,γ ;γ,ν +ν +γ; ), and E(X X ) = ν ν Γ(ν +ν +γ α)γ(ν +ν +γ β) (γ )(γ ) Γ(ν +ν +γ)γ(ν +ν +γ α β) 3 F (α,β,γ ;γ,ν +ν +γ; ). Using E(X i ), E(X i ) and E(X X ), the expressions for Var(X i ), Cov(X, X ) and Corr(X, X ) can easily be calculated. The stress-strength model describes the life of a component which has a random strength X and is subjected to a random stress X. The component fails at the instant that the stress applied to it exceeds the strength and the component will function satisfactorily whenever X > X. Thus, R= Pr(X < X ) is a measure of the component reliability. In a recent paper, Nadrajah[7] has give an extensive survey on applications and computation of R when X and X follows bivariate distribution with dependence between them. If(X, X ) has a bivariate inverted hypergeometric function type I distribution, then ν R= C(ν,ν ;α,β,γ) x ν x ν +ν x + x + x +γ F α,β;γ; dx dx. + x + x

5 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), Replacing F α,β;γ;(+ x + x ) by its series representation, we get R= C(ν,ν ;α,β,γ) Now, using (A.8), we have i= (α) i (β) i (γ) i i! ν x ν x ν +ν x + x + x +γ+i dx dx. R = C(ν,ν ;α,β,γ) i= (α) i (β) i (ν +γ+ i)(γ) i i! x (γ+i+) F ν +ν +γ+ i,ν +γ+i;ν +γ+i+ ; + x x dx. Finally, using (A.6), expanding F in series form and using (A.7), we obtain R = C(ν,ν ;α,β,γ) i= k= (α) i (β) i (ν +ν +γ+ i) k (ν +γ+ i)(γ) i (ν +γ+i+ ) k i! Γ(ν +ν )Γ(γ+ i) Γ(ν +ν +γ+ i) F ν +ν +γ+ i+ k,ν +ν ;ν +ν +γ+i;. 3. Distributions of Sum and Quotients It is well known that if(x, X ) D I I (ν,ν ;ν 3 ), then X /X and X /(X + X ) are independent of X + X. Further, X /X B I I (ν,ν ), X /(X + X ) B I (ν,ν ), and X + X B I I (ν +ν,ν 3 ). In this section we derive similar results when X and X have a bivariate inverted hypergeometric function type I distribution. Theorem. Let(X, X ) IH I (ν,ν ;α,β,γ). Then, Z= X /(X + X ) and S=X + X are independent, Z B I (ν,ν ) and S IH I (ν +ν,α,β,γ). Proof. Transforming Z= X /(X + X ) and S=X + X with the Jacobian J(x, x z,s)=s in (), we obtain the joint p.d.f. of Z and S as C(ν,ν ;α,β,γ)z ν ( z) ν s ν +ν (+s) ν +ν +γ F α,β;γ;, +s where <z< and s>. Now, from the above factorization it is clear that Z and S are independent, Z B I (ν,ν ) and S IH I (ν +ν,α,β,γ). Corollary. Let(X, X ) IH I (ν,ν ;α,β,γ). Then, X /X and X + X are independent. Further, X /X B I I (ν,ν ).

6 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), Products of Two Independent Random Variables Let(X, X ) and X 3 be independent,(x, X ) IH I (ν,ν ;α,β,γ). In this section we derive density functions of(x X 3, X X 3 ) when (i) X 3 IH I (κ,µ,ρ,σ), (ii) X 3 B I I (κ,σ), (iii) X 3 KB(κ,µ,λ), (iv) X 3 B I (κ,µ), (v) X 3 B I I I (κ,µ), and (vi) X 3 H I (κ,µ,ρ,σ). Throughout this section we writeν +ν =ν. Theorem 3. Let(X, X ) and X 3 be independent,(x, X ) IH I (ν,ν ;α,β,γ), and X 3 IH I (κ,µ,ρ,σ). Then, the p.d.f. of(z, Z )=(X, X )X 3 is given by Γ(ν+γ α)γ(ν+γ β) Γ(σ+κ µ)γ(σ+κ ρ) Γ(ν+σ)Γ(κ+γ) Γ(ν )Γ(ν )Γ(γ)Γ(ν+γ α β) Γ(σ)Γ(κ)Γ(σ+κ µ ρ) Γ(ν+γ+κ+σ) z ν z ν (α) r (µ) s (β) r (ρ) s (κ+γ) r (ν+σ) s (γ) r (σ) s (ν+κ+γ+σ) r+s r!s! where z > and z >. r= s= F (ν+σ+s,ν+γ+ r;ν+κ+γ+σ+ r+ s; z z ), (6) Proof. Using independence, the joint p.d.f. of (X,X ) and X 3 is given by K x ν x ν x κ 3 (+ x + x ) ν+γ (+ x 3 ) κ+σ F α,β;γ; where x >, x >, x 3 > and F µ,ρ;σ;, + x + x + x 3 Γ(ν+γ α)γ(ν+γ β) Γ(σ+κ µ)γ(σ+κ ρ) K = Γ(ν )Γ(ν )Γ(γ)Γ(ν+γ α β) Γ(σ)Γ(κ)Γ(σ+κ µ ρ). Transforming Z = X X 3, Z = X X 3 U= /(+X 3 ) with the Jacobian J(x, x, x 3 z,z,u)=/( u) in the joint density of(x, X ) and X and integrating u, we obtain the p.d.f. of(z, Z ) as K z ν z ν u ν+σ ( u) κ+γ u [ ( z z )u] ν+γ F α,β;γ; ( z z )u F µ,ρ;σ; u du. (7)

7 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), Now, expanding Gauss hypergeometric functions in the integral (7) in terms of power series we arrive at K z ν z ν r= s= (α) r (µ) s (β) r (ρ) s (γ) r (σ) s r!s! u ν+σ+s ( u) κ+γ+r [ ( z z )u] Finally, using (A.5) and substituting for K we obtain the desired result. ν+γ+r du. Corollary. Let(X, X ) and X 3 be independent,(x, X ) IH I (ν,ν ;α,β,γ), and X 3 B I I (κ,σ). Then, the p.d.f of(z, Z )=(X, X )X 3 is given by Γ(γ+ν α)γ(γ+ν β) Γ(κ+σ) Γ(ν+σ)Γ(κ+γ) Γ(γ)Γ(ν )Γ(ν )Γ(γ+ν α β) Γ(σ)Γ(κ) Γ(ν+κ+γ+σ) zν z ν (α) r (β) r (κ+γ) r (γ) r (ν+κ+γ+σ) r r! r= where z > and z >. F (ν+σ,ν+γ+ r;ν+κ+γ+σ+ r; z z ), (8) Corollary 3. Let(X, X ) and X 3 be independent,(x, X ) D I I (ν,ν ;γ), and X 3 B I I (κ,σ). Then, the p.d.f of(z, Z )=(X, X )X 3 is given by where z > and z >. Γ(γ+ν) Γ(κ+σ) Γ(ν+σ)Γ(κ+γ) Γ(γ)Γ(ν )Γ(ν ) Γ(σ)Γ(κ) Γ(ν+κ+γ+σ) z ν z ν F (ν+σ,ν+γ;ν+κ+γ+σ; z z ), (9) Note that the Gauss hypergeometric functions in the densities (6), (8) and (9) can be expanded in series form if <z +z <. However, if z +z >, then /(z +z )<and we use (A.6) to rewrite the densities (6), (8) and (9) in series involving Gauss hypergeometric functions having /(z + z ) as argument. The next theorem gives the density of the product of Kummer-beta and inverted hypergeometric function type I variables. Theorem 4. Let(X, X ) and X 3 be independent,(x, X ) IH I (ν,ν ;α,β,γ) and X 3 KB(κ,µ,λ). Then, the p.d.f. of(z, Z )=(X, X )X 3 is Γ(ν+γ α)γ(ν+γ β)γ(µ+κ)γ(γ+κ) Γ(ν )Γ(ν )Γ(κ)Γ(γ)Γ(ν+γ α β)γ(γ+µ+κ) { F (µ;κ+µ;λ)} zν z ν (+z + z ) ν+γ r= Φ µ,ν+γ+ r;γ+µ+κ+ r; (α) r (β) r (γ+κ) r (γ+µ+κ) r (γ) r r! (+z + z ) r,λ, z >, z >. +z + z

8 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), Proof. The joint p.d.f. of(x, X ) and X 3 is given by x ν x ν x κ 3 ( x 3 ) µ K (+ x + x ) ν+γ F α,β;γ; where x >, x >, < x < and K = + x + x exp[λ( x 3 )], () Γ(ν+γ α)γ(ν+γ β) Γ(ν )Γ(ν )Γ(γ)Γ(ν+γ α β) {B(κ,µ) F (µ;κ+µ;λ)}. Transforming Z = X X 3, Z = X X 3 and W= X 3 with the Jacobian J(x, x, x 3 z,z, w)=/( w) in () and integrating w, we obtain the joint p.d.f. of Z and Z as K z ν z ν (+z + z ) ν+γ exp(λw) F w µ ( w) γ+κ w/(+z + z ) ν+γ α,β;γ; (+z + z ) ( w) w/(+z + z ) dw. () Now, expanding Gauss hypergeometric functions in the integral () in terms of power series we arrive at K z ν z ν (+z + z ) ν+γ r= (α) r (β) r (γ) r r! (+z + z ) r w µ ( w) γ+κ+r exp(λw) w/(+z + z ) dw. ν+γ+r Finally, applying (A.) and substituting for K we obtain the desired result. Corollary 4. Let(X, X ) and X 3 be independent,(x, X ) D II (ν,ν ;γ) and X 3 KB(κ,µ,λ). Then, the p.d.f. of(z, Z )=(X, X )X 3 is given by Γ(γ+ν)Γ(µ+κ)Γ(γ+κ) Γ(ν )Γ(ν )Γ(κ)Γ(γ)Γ(γ+µ+κ) { F (µ;κ+µ;λ)} zν z ν (+z + z ) ν+γφ µ,ν+γ;γ+µ+κ;,λ, z >, z >. +z + z Corollary 5. Let(X, X ) and X 3 be independent,(x, X ) D I I (ν,ν ;γ) and X 3 B I (κ,µ). Then, the p.d.f. of(z, Z )=(X, X )X 3 is given by Γ(γ+ν)Γ(µ+κ)Γ(γ+κ) z ν z ν Γ(ν )Γ(ν )Γ(κ)Γ(γ)Γ(γ+µ+κ)(+z + z ) ν+γ F µ,ν+γ;γ+µ+κ;, z >, z >. +z + z

9 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), Corollary 6. Let(X, X ) and X 3 be independent,(x, X ) IH I (ν,ν ;α,β,γ) and X 3 B I (κ,µ). Then, the p.d.f. of(z, Z )=(X, X )X 3 is given by Γ(ν+γ α)γ(ν+γ β)γ(µ+κ)γ(γ+κ) Γ(ν )Γ(ν )Γ(κ)Γ(γ)Γ(ν+γ α β)γ(γ+µ+κ) zν z ν (α) r (β) r (γ+κ) r (+z + z ) ν+γ (γ+µ+κ) r= r (γ) r r! (+z + z ) r F µ,ν+γ+ r;γ+µ+κ+ r;, z >, z >. +z + z Theorem 5. Let(X, X ) and X 3 be independent,(x, X ) IH I (ν,ν ;α,β,γ) and X 3 B I I I (κ,µ). Then, the p.d.f. of(z, Z )=(X, X )X 3 is given by where z > and z >. Γ(ν+γ α)γ(ν+γ β)γ(κ+µ)γ(κ+γ) µ Γ(ν )Γ(ν )Γ(κ)Γ(γ)Γ(ν+γ α β)γ(κ+µ+γ) zν z ν (α) r (β) r (γ+κ) r (+z + z ) ν+γ (γ) r= r (γ+κ+µ) r r! (+z + z ) r F µ;ν+γ+ r,µ+κ;γ+κ+µ+ r;,, +z + z Proof. The joint p.d.f. of(x, X ) and X 3 is given by x ν x ν x κ 3 ( x 3 ) µ K 3 (+ x + x ) ν+γ (+ x 3 ) κ+µ F α,β;γ;, () + x + x where x >, x >, < x 3 < and K 3 = Γ(ν+γ α)γ(ν+γ β) Γ(ν )Γ(ν )Γ(γ)Γ(ν+γ α β) κ {B(κ,µ)}. Now, transforming Z = X X 3, Z = X X 3 and W= X 3 with the Jacobian J(x, x, x 3 z,z, w)=/( w) in () and integrating w, the marginal p.d.f. of(z, Z ) is derived as K 3 z ν z ν κ+µ (+z + z ) ν+γ F w µ ( w) κ+γ w/(+z + z ) ν+γ ( w/) κ+µ α,β;γ; (+z + z ) ( w) w/(+z + z ) dw. (3) Expanding Gauss hypergeometric functions in the integral (3) in series form we arrive at K 3 z ν z ν κ+µ (+z + z ) ν+γ (α) r (β) r (γ) r r!(+z + z ) r r= w µ ( w) κ+γ+r w/(+z + z ) ν+γ+r ( w/) κ+µ dw. Finally, the desired result follows by using (A.) and substituting for K 3.

10 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), Corollary 7. Let(X, X ) and X 3 be independent,(x, X ) D I I (ν,ν ;γ) and X 3 B I I I (κ,µ). Then, the p.d.f. of(z, Z )=(X, X )X 3 is given by where z > and z >. Γ(ν+γ)Γ(κ+µ)Γ(κ+γ) z ν z ν µ Γ(ν )Γ(ν )Γ(κ)Γ(γ)Γ(κ+µ+γ)(+z + z ) ν+γ F µ;ν+γ,µ+κ;γ+κ+µ;,, +z + z Theorem 6. Let(X, X ) and X 3 be independent,(x, X ) IH I (ν,ν ;α,β,γ) and X 3 H I (κ,µ,ρ,σ). Then, the p.d.f. of(z, Z )=(X, X )X 3 is given by where z > and z >. Γ(ν+γ α)γ(ν+γ β)γ(κ+γ)γ(σ+κ µ)γ(σ+κ ρ) Γ(ν )Γ(ν )Γ(κ)Γ(γ)Γ(ν+γ α β)γ(σ+κ µ ρ)γ(κ+γ+σ) zν z ν (µ) s (ρ) s (α) r (β) r (κ+γ) r (+z + z ) ν+γ (γ) s= r= r (κ+γ+σ) s+r s! r! (+z + z ) r F σ+s,ν+γ+ r;κ+σ+γ+s+r;, +z + z Proof. The joint p.d.f. of(x, X ) and X 3 is given by K 4 x ν x ν x κ 3 ( x 3 ) σ (+ x + x ) ν+γ F α,β;γ; where x >, x >, < x 3 < and + x + x F (µ,ρ;σ; x 3 ), (4) Γ(ν+γ α)γ(ν+γ β) Γ(σ+κ µ)γ(σ+κ ρ) K 4 = Γ(ν )Γ(ν )Γ(γ)Γ(ν+γ α β) Γ(σ)Γ(κ)Γ(σ+κ µ ρ). Now, transforming Z = X X 3, Z = X X 3 and W= X 3 with the Jacobian J(x, x, x 3 z,z, w)=/( w) in (4) and integrating w, we obtain the p.d.f. of (Z, Z ) as K 4 z ν z ν w σ ( w) κ+γ (+z + z ) ν+γ w/(+z + z ) ν+γ F α,β;γ; (+z + z ) ( w) F µ,ρ;σ; w dw, z >, z >. w/(+z + z ) Now, expanding the Gauss hypergeometric functions in series form, integrating the resulting expression using (A.5), substituting for K 4 and simplifying, we obtain the desired result.

11 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), Corollary 8. Let(X, X ) and X 3 be independent,(x, X ) D I I (ν,ν ;γ) and X 3 H I (κ,µ,ρ,σ). Then, the p.d.f. of(z, Z )=(X, X )X 3 is given by Γ(ν+γ)Γ(κ+γ)Γ(σ+κ µ)γ(σ+κ ρ) z ν z ν Γ(ν )Γ(ν )Γ(κ)Γ(γ)Γ(σ+κ µ ρ)γ(κ+γ+σ)(+z + z ) ν+γ (µ) s (ρ) s (κ+γ+σ) s s! F σ+s,ν+γ;κ+σ+γ+s;, +z + z s= where z > and z >. Theorem 7. Let(X, X ), Y and Y be independent,(x, X ) IH I (ν,ν ;α,β,γ) and Y i B I (a i, b i ), i=,. Then, the p.d.f of(z, Z )=(X, X )Y Y is given by Γ(a + b )Γ(a + b )Γ(a +γ) Γ(γ+ν α)γ(γ+ν β) Γ(a )Γ(a )Γ(a + b + b +γ) Γ(ν )Γ(ν )Γ(γ)Γ(γ+ν α β) zν z ν (b ) s (a + b a ) s (α) r (β) r (a +γ) r (+z + z ) ν+γ (+z + z ) r (γ) r= s= r (a + b + b +γ) s+r s! r! F b + b + s,ν+γ+ r; a + b + b +γ+s+r;, +z + z where z > and z >. Proof. Using Theorem A.8, Y Y H I (a, b, a + b a, b + b ). Now, using independence of(x, X ) and X 3 and Theorem 6, we obtain the desired result. Corollary 9. Let(X, X ), Y and Y be independent,(x, X ) D I I (ν,ν ;γ) and Y i B I (a i, b i ), i=,. Then, the p.d.f of(z, Z )=(X, X )Y Y is given by Γ(a + b )Γ(a + b )Γ(a +γ)γ(ν+γ) z ν z ν Γ(a )Γ(a )Γ(ν )Γ(ν )Γ(γ)Γ(a + b + b +γ) (+z + z ) ν+γ (b ) s (a + b a ) s (a + b + b +γ) s s! F b + b + s,ν+γ; a + b + b +γ+s;, +z + z s= where z > and z >. Appendix: Some Known Definitions and Results Here, we give some definitions and additional results which are used throughout this work. We use the Pochhammer symbol(a) n defined by (a) n = a(a+) (a+n )=(a) n (a+n ) for n=,,...,(a) =.

12 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), The generalized hypergeometric function of scalar argument is defined by pf q (a,...,a p ; b,..., b q ; z)= k= (a ) k (a p ) k z k (b ) k (b q ) k k!, (A.) 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[5]. From (A.) it is easy to see that and F (z)= k= z k k! = exp(z), F (a; c; z)= F (a, b; c; z)= k= (a) k (b) k z k (c) k k!, z <. Also, under suitable conditions, we have from Luke[5, Eq. 3.6()], k= (a) k z k (c) k k!, z α ( z) β pf q (a,..., a p ; b,..., b q ; z y) dz = Γ(α)Γ(β) Γ(α+β) p+ F q+ (a,..., a p,α; b,..., b q,α+β; y) (A.) and Luke[5, Eq. 3.6(3)], exp( δz)z α pf q (a,..., a p ; b,..., b q ; z y) dz =Γ(α)δ α p+f q (a,..., a p,α; b,..., b q ;δ y). (A.3) The integral representations of the confluent hypergeometric function and the Gauss hypergeometric function are given as Γ(c) F (a; c; z)= Γ(a)Γ(c a) t a ( t) c a exp(zt) dt, (A.4) and Γ(c) F (a, b; c; z)= Γ(a)Γ(c a) t a ( t) c a ( zt) b dt, (A.5) respectively, where Re(a)> and Re(c a)>. It is easy to check by using (A.5) that F (a, b; c; z) = ( z) a z F a, c b; c; z = ( z) b z F c a, b; c; (A.6) z

13 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), and for Re(a)>, Re(b+ c)> and arg(z) <π, it has been shown that[see Luke 5, Eq ], Γ(a+ b+ c) F (a, b; a+ b+ c; z)= Γ(b)Γ(a+ c) Further, for Re(λ)<Re(ν), we have[prudnikov, Eq...4.4], x y λ xλ ν dy= (y+ a) ν ν λ F The Appell s first hypergeometric function F is defined by F (a; b, b ; c; z,z ) = = = r,s= r= s= s b (+s) c b ds (+sz) a. (A.7) ν,ν λ; +ν λ; a. (A.8) x (a) r+s (b ) r (b ) s (c) r+s z r zs r!s! (a) r (b ) r (c) r z r r! F (a+ r, b ; c+r; z ) (a) s (b ) s (c) s z s s! F (a+ s, b ; c+ s; z ), (A.9) where z < and z <. The Humbert s confluent hypergeometric functionφ is defined by Φ [a, b ; c; z,z ] = = = r,s= r= s= (a) r+s (b ) r (c) r+s z r zs r!s!, (a) r (b ) r (c) r z r r! F (a+ r; c+r; z ) (a) s (c) s z s s! F (a+s, b ; c+ s; z ), (A.) where z <, z <. The integral representations of F andφ are given by and Γ(c) F (a; b, b ; c; z,z )= Γ(a)Γ(c a) Γ(c) Φ [a, b ; c; z,z ]= Γ(a)Γ(c a) v a ( v) c a dv ( vz ) b ( vz ) b, (A.) v a ( v) c a exp(vz ) dv ( vz ) b, (A.) where Re(a)>and Re(c a)>. Note that for b =, F andφ reduce to F and F functions, respectively. For properties and further results on these functions the reader is referred to Luke[5] and Srivastava and Karlsson[5]. Next, we define the beta type I, beta type II, beta type III, hypergeometric function type I and Kummer-beta distributions. These definitions can be found in Gordy[], Johnson, Kotz and Balakrishnan[4], Nagar and Zarrazola[], and Sánchez and Nagar[3].

14 Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar/Eur. J. Pure Appl. Math, 5 (), Definition A.. The random variable X is said to have a beta type I distribution with parameters (a, b), a>, b>, denoted as X B I (a, b), if its p.d.f. is given by where B(a, b) is the beta function given by {B(a, b)} x a ( x) b, < x<, B(a, b)=γ(a)γ(b){γ(a+ b)}. Definition A.. The random variable X is said to have a beta type II distribution with parameters (a, b), denoted as X B I I (a, b), a>, b>, if its p.d.f. is given by {B(a, b)} x a (+ x) (a+b), x>. Definition A.3. The random variable X is said to have a beta type III distribution with parameters(a, b), denoted as X B I I I (a, b), a>, b>, if its p.d.f. is given by a {B(a, b)} x a ( x) b (+ x) (a+b), < x<. Definition A.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 whereα>,β> and <λ<. x α ( x) β exp[λ( x)], < x<, B(α,β) F (β;α+β;λ) Note that forλ= the above density simplifies to a beta type I density with parametersα andβ. The bivariate generalizations of beta type I and beta type II distributions are defined next. Definition A.5. The random variables X and Y are said to have a Dirichlet type I distribution of order 3 with parameters(a, b, c), a>, b>, c>, denoted as X D I (a, b; c), if their joint p.d.f. is given by {B(a, b, c)} x a y b ( x y) c, x>, y>, x+ y<, where B(a, b, c) is defined by B(a, b, c)=γ(a)γ(b)γ(c){γ(a+ b+c)}. Definition A.6. The random variables X and Y are said tohave a Dirichlet type II distribution of order 3 with parameters(a, b, c), a>, b>, c>, denoted as X D I I (a, b; c), if their joint p.d.f. is given by {B(a, b, c)} x a y b (+ x+ y) (a+b+c), x>, y>.

15 REFERENCES 33 Definition A.7. 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 α)γ(γ+ν β) Γ(γ)Γ(ν)Γ(γ+ν α β) xν ( x) γ F (α,β;γ; x), < x<, whereγ+ν α β>,γ> andν>. The following result (Gupta and Nagar[3], Nagar and Alvarez[8]) 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 A.8. Let X and X be independent, X i B I (a i, b i ), i=,. Then, X X H I (a, b, a + b a, b + b ). 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[], Gupta and Nagar[3], and Nagar and Gupta[]. Acknowledgments The research work of DKN was supported by the Comité para el Desarrollo de la Investigación, Universidad de Antioquia research grant no. IN56CE. References [] M. B. Gordy. Computationally convenient distributional assumptions for common-value auctions. Comput. Econom., :6 78, 998. [] A. K. Gupta and D. K. Nagar. Matrix variate beta distribution. Int. J. Math. Math. Sci., 4(7): ,. [3] A. K. Gupta and D. K. Nagar. Matrix Variate Distributions, volume 4 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton,. [4] N. L. Johnson, S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Vol.. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons Inc., New York, 995. [5] Y. L. Luke. The Special Functions and Their Approximations. Vol. I, volume 53 of Mathematics in Science and Engineering. Academic Press, New York, 969. [6] A. W. Marshall and I. Olkin. Inequalities: theory of majorization and its applications, volume 43 of Mathematics in Science and Engineering. Academic Press Inc.[Harcourt Brace Jovanovich Publishers], New York, 979. [7] S. Nadarajah. Reliability for some bivariate beta distributions. Math. Probl. Eng., ():, 5.

16 REFERENCES 33 [8] D. K. Nagar and J. A. Alvarez. Properties of the hypergeometric function type I distribution. Adv. Appl. Stat., 5(3):34 35, 5. [9] D. K. Nagar, P. A. Bran-Cardona, and A. K. Gupta. Multivariate generalization of the hypergeometric function type I distribution. Acta Appl. Math., 5():, 9. [] D. K. Nagar and A. K. Gupta. Matrix-variate Kummer-beta distribution. J. Aust. Math. Soc., 73(): 5,. [] D. K. Nagar and E. Zarrazola. Distributions of the product and the quotient of independent Kummer-beta variables. Sci. Math. Jpn., 6():9 7, 5. [] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev. Integrals and series. Vol.. Gordon & Breach Science Publishers, New York, 986. Elementary functions, Translated from the Russian and with a preface by N. M. Queen. [3] L. E. Sánchez and D. K. Nagar. Distributions of the product and the quotient of independent beta type 3 variables. Far East J. Theor. Stat., 7():39 5, 5. [4] B. D. Sivazlian. On a multivariate extension of the gamma and beta distributions. SIAM J. Appl. Math., 4():5 9, 98. [5] H. M. Srivastava and P. W. Karlsson. Multiple Gaussian Hypergeometric Series. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester, 985. [6] E. Zarrazola and D. K. Nagar. Product of independent inverted hypergeometric function type I variables. Revista Ingeniería y Ciencia, Universidad Eafit, 5():93 6, 9.