4_nagar.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 3, 2012, 317-332

ISSN 1307-5543 – www.ejpam.com

Bivariate Generalization of The Inverted Hypergeometric

Function Type I Distribution

Paula A. Bran-Cardona1, Edwin Zarrazola2 and Daya K. Nagar 2,∗

1 Departamento de Matemáticas, Universidad del Valle, Calle 13, No. 100-00, Cali, Colombia
2 Instituto de Matemáticas, Universidad de Antioquia, Calle 67, No. 53-108, Medellín, Colombia

Abstract. The bivariate inverted hypergeometric function type I distribution is defined by the proba-

bility density function proportional to x
ν1−1

1 x
ν2−1

2

�

1+ x1+ x2

�−(ν1+ν2+γ)
2F1(α,β ;γ; (1+ x1 + x2)

−1),

x1 > 0, x2 > 0, where ν1, ν2, α, β and γ are suitable constants. In this article, we study several

properties of this distribution and derive density functions of X1/X2, X1/(X1 + X2) and X1 + X2. 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.

2010 Mathematics Subject Classifications: 62E15, 60E05

Key Words and Phrases: Appell’s first hypergeometric function, Beta distribution, Gauss hypergeo-

metric function, Humbert’s confluent hypergeometric function, product, transformation.

1. Introduction

The random variable X is said to have an inverted hypergeometric function type I distri-

bution, 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)

Γ(γ)Γ(ν)Γ(γ+ ν −α− β)

xν−1

(1+ x)ν+γ
2F1

�

α,β ;γ;
1

1+ x

�

, x > 0, (1)

where ν > 0, γ > 0, γ+ ν > α+ β , and 2F1 is the Gauss hypergeometric function. For α = γ,

the density (1) reduces to a beta type II density given by

Γ(γ+ ν − β)

Γ(γ)Γ(ν − β)

xν−β−1

(1+ x)ν−β+γ
, x > 0,

∗Corresponding author.

Email addresses: paula.bran�gmail.
om (Bran-Cardona), ezarrazo�gmail.
om (Zarrazola),

dayaknagar�yahoo.
om (Nagar)

http://www.ejpam.com 317 c© 2012 EJPAM All rights reserved.



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 318

and for β = γ, the inverted hypergeometric function type I density slides to

Γ(γ+ ν −α)

Γ(γ)Γ(ν −α)

xν−α−1

(1+ x)ν−α+γ
, x > 0.

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 Alvarez [8] studied several properties and stochastic representations

of the inverted hypergeometric function type I distribution. Zarrazola and Nagar [16] derived

the density function of the product of two independent random variables having inverted hy-

pergeometric 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 (X1, X2)∼ IH I (ν1,ν2;α,β ,γ), is defined by the density [Nagar, Bran-Cardona and

Gupta 9],

C(ν1,ν2;α,β ,γ)
x
ν1−1

1
x
ν2−1

2
�

1+ x1+ x2

�ν1+ν2+γ 2F1

�

α,β ;γ;
1

1+ x1+ x2

�

, (2)

where x1 > 0, x2 > 0, and C(ν1,ν2;α,β ,γ) is the normalizing constant given by

C(ν1,ν2;α,β ,γ) =
Γ(ν1 + ν2 + γ−α)Γ(ν1 + ν2 + γ− β)

Γ(ν1)Γ(ν2)Γ(γ)Γ(ν1+ ν2 + γ−α− β)
,

with ν1 > 0,ν2 > 0, γ > 0, and ν1 + ν2 + γ > α+ β .

For α = 0 or β = 0, the density (2) slides to a Dirichlet type II density of order 3 with

parameters ν1, ν2 and γ.

It can also be observed that bivariate generalization of the hypergeometric function type I

distribution defined by the density (2) belongs to the Liouville family of distributions proposed

by Marshall and Olkin [6] and Sivazlian [14].

In this article we study several properties of the bivariate generalization of the hypergeo-

metric function type I distribution defined by the density (2).

In Section 2, we derive results such as the marginal and the conditional densities, moments

and correlation and in Section 3 we show that if (X1, X2)∼ IH I (ν1,ν2;α,β ,γ), then,

X1+ X2 ∼ H I (ν1 + ν2,α,β ,γ), which is independent of X1/(X1+ X2)∼ BI (ν1,ν2) and

X1/X2 ∼ BI I (ν1,ν2). In Section 4, we derive density functions of (X1X3, X2X3), where (X1, X2)

and X3 are independent, (X1, X2)∼ IH I(ν1,ν2;α,β ,γ) and

(i) X3 ∼ IH I (κ,µ,ρ,σ),

(ii) X3 ∼ BI I(κ,σ),

(iii) X3 ∼ KB(κ,µ,λ)

(iv) X3 ∼ BI (κ,µ),



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 319

(v) X3 ∼ BI I I(κ,µ), and

(vi) X3 ∼ H I(κ,µ,ρ,σ).

Finally, in appendix we give definitions and results on Gauss hypergeometric function, Ap-

pell’s first hypergeometric function F1, Humbert’s confluent hypergeometric function Φ1 and

statistical distributions.

2. Properties

In this section we study several properties of the bivariate distribution defined in Section 1.

We first derive marginal and conditional distributions.

Theorem 1. If (X1, X2)∼ IH I (ν1,ν2;α,β ,γ), then the p.d.f. of X1 is given by

Γ(ν1 + γ)Γ(ν1 + ν2 + γ−α)Γ(ν1 + ν2 + γ− β)

Γ(ν1)Γ(γ)Γ(ν1 + ν2 + γ)Γ(ν1 + ν2 + γ−α− β)

×
x
ν1−1

1

(1+ x1)
ν1+γ 3F2

�

α,β ,ν1 + γ;γ,ν1 + ν2 + γ;
1

1+ x1

�

, x1 > 0. (3)

Proof. By integrating x2 in (2), we get the marginal p.d.f. of X1 as

C(ν1,ν2;α,β ,γ)
x
ν1−1

1

(1+ x1)
ν1+γ

∫ 1

0

zν1+γ−1(1− z)ν2−1
2F1

�

α,β ;γ;
z

1+ x1

�

dz,

where we have used the substitution z = (1+ x1)/(1+ x1 + x2). Now, the desired result is

obtained by using (A.2).

For α = γ, the density (2) reduces to

Γ(ν1 + ν2)Γ(ν1 + ν2 + γ− β)

Γ(ν1)Γ(ν2)Γ(γ)Γ(ν1 + ν2 − β)

x
ν1−1

1 x
ν2−1

2
�

x1+ x2

�β �
1+ x1 + x2

�ν1+ν2+γ−β
, (4)

where x1 > 0 and x2 > 0. The marginal density of X1 in this case is given by

Γ(ν1 + γ)Γ(ν1 + ν2)Γ(ν1 + ν2 + γ− β)

Γ(ν1)Γ(γ)Γ(ν1+ ν2 + γ)Γ(ν1 + ν2 − β)

×
x
ν1−1

1

(1+ x1)
ν1+γ 2F1

�

β ,ν1 + γ;ν1 + ν2 + γ;
1

1+ x1

�

, x1 > 0. (5)

Using the above theorem, the conditional density function of X1 given X2 = x2 > 0 is obtained

as

Γ(γ+ ν1 + ν2)

Γ(ν1)Γ(γ+ ν2)

x
ν1−1

1
(1+ x2)

γ+ν2

(1+ x1+ x2)
γ+ν1+ν2

2F1(α,β ;γ; (1+ x1+ x2)
−1)

3F2(α,β ,γ+ ν2;γ,γ+ ν1 + ν2; (1+ x2)
−1)

,



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 320

where x1 > 0 and x2 > 0. Further, using (2), the joint (r, s)-th moment is obtained as

E(X r
1X s

2) = C(ν1,ν2;α,β ,γ)

∫ ∞

0

∫ ∞

0

x
ν1+r−1

1
x
ν2+s−1

2

(1+ x1+ x2)
ν1+ν2+γ

×2F1

�

α,β ;γ;
1

1+ x1 + x2

�

dx2 dx1.

Now, substituting u= x1/(x1 + x2), v = x1+ x2 and z = 1/(1+ v) with the Jacobian

J(x1, x2→ u, z) = J(x1, x2→ u, v)J(v→ z) = (1− z)/z3 in the above integral, one obtains

E(X r
1X s

2) = C(ν1,ν2;α,β ,γ)B(ν1 + r,ν2 + s)

∫ 1

0

zγ−r−s−1(1− z)ν1+ν2+r+s−1
2F1(α,β ;γ; z)dz.

Finally, evaluating the above integral using (A.2) and simplifying the resulting expression, we

get

E(X r
1X s

2) =
Γ(ν1 + r)Γ(ν2 + s)Γ(γ− r − s)Γ(ν1 + ν2 + γ−α)Γ(ν1 + ν2 + γ− β)

Γ(ν1)Γ(ν2)Γ(γ)Γ(ν1 + ν2 + γ)Γ(ν1 + ν2 + γ−α− β)
×3F2(α,β ,γ− r − s;γ,ν1 + ν2 + γ; 1),

where ν1 + r > 0, ν2 + s > 0 and γ > r + s. Now, substituting appropriately, we obtain

E(X i) =
νi

γ− 1

Γ(ν1 + ν2 + γ−α)Γ(ν1 + ν2 + γ− β)

Γ(ν1 + ν2 + γ)Γ(ν1 + ν2 + γ−α− β)
×3F2(α,β ,γ− 1;γ,ν1 + ν2 + γ; 1),

E(X 2
i ) =

νi(νi + 1)

(γ− 1)(γ− 2)

Γ(ν1 + ν2 + γ−α)Γ(ν1 + ν2 + γ− β)

Γ(ν1 + ν2 + γ)Γ(ν1 + ν2 + γ−α− β)
×3F2(α,β ,γ− 2;γ,ν1 + ν2 + γ; 1),

and

E(X1X2) =
ν1ν2

(γ− 1)(γ− 2)

Γ(ν1 + ν2 + γ−α)Γ(ν1 + ν2 + γ− β)

Γ(ν1 + ν2 + γ)Γ(ν1 + ν2 + γ−α− β)
×3F2(α,β ,γ− 2;γ,ν1 + ν2 + γ; 1).

Using E(X i), E(X 2
i ) and E(X1X2), the expressions for Var(X i), Cov(X1, X2) and Corr(X1, X2)

can easily be calculated.

The stress-strength model describes the life of a component which has a random strength

X2 and is subjected to a random stress X1. The component fails at the instant that the stress

applied to it exceeds the strength and the component will function satisfactorily whenever

X2 > X1. Thus, R = Pr(X1 < X2) 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 X1 and

X2 follows bivariate distribution with dependence between them. If (X1, X2) has a bivariate

inverted hypergeometric function type I distribution, then

R= C(ν1,ν2;α,β ,γ)

∫ ∞

0

x
ν1−1

1

∫ ∞

x1

x
ν2−1

2
�

1+ x1+ x2

�ν1+ν2+γ 2F1

�

α,β ;γ;
1

1+ x1+ x2

�

dx2 dx1.



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 321

Replacing 2F1

�

α,β ;γ; (1+ x1+ x2)
−1
�

by its series representation, we get

R= C(ν1,ν2;α,β ,γ)

∞
∑

i=0

(α)i(β)i

(γ)i i!

∫ ∞

0

x
ν1−1

1

∫ ∞

x1

x
ν2−1

2
�

1+ x1+ x2

�ν1+ν2+γ+i
dx2 dx1.

Now, using (A.8), we have

R = C(ν1,ν2;α,β ,γ)

∞
∑

i=0

(α)i(β)i

(ν1 + γ+ i) (γ)i i!

×

∫ ∞

0

x
−(γ+i+1)

1 2F1

�

ν1 + ν2 + γ+ i,ν1 + γ+ i;ν1 + γ+ i+ 1;−
1+ x1

x1

�

dx1.

Finally, using (A.6), expanding 2F1 in series form and using (A.7), we obtain

R = C(ν1,ν2;α,β ,γ)

∞
∑

i=0

∞
∑

k=0

(α)i(β)i (ν1 + ν2 + γ+ i)k

(ν1 + γ+ i)(γ)i(ν1 + γ+ i+ 1)k i!

×
Γ(ν1 + ν2)Γ(γ+ i)

Γ(ν1+ ν2 + γ+ i)
2F1

�

ν1 + ν2 + γ+ i + k,ν1 + ν2;ν1 + ν2 + γ+ i;−1
�

.

3. Distributions of Sum and Quotients

It is well known that if (X1, X2) ∼ DI I(ν1,ν2;ν3), then X1/X2 and X1/(X1 + X2) are

independent of X1 + X2. Further, X1/X2 ∼ BI I(ν1,ν2), X1/(X1 + X2) ∼ BI (ν1,ν2), and

X1 + X2 ∼ BI I (ν1 + ν2,ν3). In this section we derive similar results when X1 and X2 have

a bivariate inverted hypergeometric function type I distribution.

Theorem 2. Let (X1, X2) ∼ IH I (ν1,ν2;α,β ,γ). Then, Z = X1/(X1 + X2) and S = X1 + X2 are

independent, Z ∼ BI (ν1,ν2) and S ∼ IH I (ν1 + ν2,α,β ,γ).

Proof. Transforming Z = X1/(X1+ X2) and S = X1 + X2 with the Jacobian

J(x1, x2→ z, s) = s in (2), we obtain the joint p.d.f. of Z and S as

C(ν1,ν2;α,β ,γ)zν1−1(1− z)ν2−1
sν1+ν2−1

(1+ s)ν1+ν2+γ 2F1

�

α,β ;γ;
1

1+ s

�

,

where 0 < z < 1 and s > 0. Now, from the above factorization it is clear that Z and S are

independent, Z ∼ BI (ν1,ν2) and S ∼ IH I(ν1 + ν2,α,β ,γ).

Corollary 1. Let (X1, X2) ∼ IH I(ν1,ν2;α,β ,γ). Then, X1/X2 and X1 + X2 are independent.

Further, X1/X2 ∼ BI I (ν1,ν2).



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 322

4. Products of Two Independent Random Variables

Let (X1, X2) and X3 be independent, (X1, X2) ∼ IH I (ν1,ν2;α,β ,γ). In this section we

derive density functions of (X1X3, X2X3) when

(i) X3 ∼ IH I (κ,µ,ρ,σ),

(ii) X3 ∼ BI I(κ,σ),

(iii) X3 ∼ KB(κ,µ,λ),

(iv) X3 ∼ BI (κ,µ),

(v) X3 ∼ BI I I(κ,µ), and

(vi) X3 ∼ H I(κ,µ,ρ,σ).

Throughout this section we write ν1 + ν2 = ν .

Theorem 3. Let (X1, X2) and X3 be independent, (X1, X2)∼ IH I (ν1,ν2;α,β ,γ), and

X3 ∼ IH I (κ,µ,ρ,σ). Then, the p.d.f. of (Z1, Z2) = (X1, X2)X3 is given by

Γ(ν + γ−α)Γ(ν + γ− β)

Γ(ν1)Γ(ν2)Γ(γ)Γ(ν + γ−α− β)

Γ(σ+ κ−µ)Γ(σ+ κ−ρ)

Γ(σ)Γ(κ)Γ(σ+ κ−µ−ρ)

Γ(ν +σ)Γ(κ+ γ)

Γ(ν + γ+ κ+σ)

×z
ν1−1

1
z
ν2−1

2

∞
∑

r=0

∞
∑

s=0

(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!

×2F1(ν +σ+ s,ν + γ+ r;ν + κ+ γ+σ+ r + s; 1− z1 − z2), (6)

where z1 > 0 and z2 > 0.

Proof. Using independence, the joint p.d.f. of (X1,X2) and X3 is given by

K1 x
ν1−1

1 x
ν2−1

2 xκ−1
3

(1+ x1+ x2)
ν+γ(1+ x3)

κ+σ 2F1

�

α,β ;γ;
1

1+ x1+ x2

�

2F1

�

µ,ρ;σ;
1

1+ x3

�

,

where x1 > 0, x2 > 0, x3 > 0 and

K1 =
Γ(ν + γ−α)Γ(ν + γ− β)

Γ(ν1)Γ(ν2)Γ(γ)Γ(ν + γ−α− β)

Γ(σ+ κ−µ)Γ(σ+ κ−ρ)

Γ(σ)Γ(κ)Γ(σ+ κ−µ−ρ)
.

Transforming Z1 = X1X3, Z2 = X2X3 U = 1/(1+ X3) with the Jacobian

J(x1, x2, x3→ z1, z2,u) = 1/(1− u)2 in the joint density of (X1, X2) and X2 and integrating u,

we obtain the p.d.f. of (Z1, Z2) as

K1z
ν1−1

1 z
ν2−1

2

∫ 1

0

uν+σ−1(1− u)κ+γ−1

[1− (1− z1 − z2)u]
ν+γ 2F1

�

α,β ;γ;
1− u

1− (1− z1 − z2)u

�

× 2F1

�

µ,ρ;σ; u
�

du. (7)



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 323

Now, expanding Gauss hypergeometric functions in the integral (7) in terms of power series

we arrive at

K1z
ν1−1

1 z
ν2−1

2

∞
∑

r=0

∞
∑

s=0

(α)r(µ)s(β)r(ρ)s

(γ)r(σ)sr!s!

∫ 1

0

uν+σ+s−1(1− u)κ+γ+r−1

[1− (1− z1 − z2)u]
ν+γ+r

du.

Finally, using (A.5) and substituting for K1 we obtain the desired result.

Corollary 2. Let (X1, X2) and X3 be independent, (X1, X2)∼ IH I (ν1,ν2;α,β ,γ), and

X3 ∼ BI I (κ,σ). Then, the p.d.f of (Z1, Z2) = (X1, X2)X3 is given by

Γ(γ+ ν −α)Γ(γ+ ν − β)

Γ(γ)Γ(ν1)Γ(ν2)Γ(γ+ ν −α− β)

Γ(κ+σ)

Γ(σ)Γ(κ)

Γ(ν +σ)Γ(κ+ γ)

Γ(ν +κ+ γ+σ)
z
ν1−1

1
z
ν2−1

2

×
∞
∑

r=0

(α)r(β)r(κ+ γ)r

(γ)r(ν + κ+ γ+σ)r r!

×2F1(ν +σ,ν + γ+ r;ν + κ+ γ+σ+ r; 1− z1 − z2), (8)

where z1 > 0 and z2 > 0.

Corollary 3. Let (X1, X2) and X3 be independent, (X1, X2)∼ DI I(ν1,ν2;γ), and X3 ∼ BI I (κ,σ).

Then, the p.d.f of (Z1, Z2) = (X1, X2)X3 is given by

Γ(γ+ ν)

Γ(γ)Γ(ν1)Γ(ν2)

Γ(κ+σ)

Γ(σ)Γ(κ)

Γ(ν +σ)Γ(κ+ γ)

Γ(ν + κ+ γ+σ)

×z
ν1−1

1 z
ν2−1

2 2F1(ν +σ,ν + γ;ν + κ+ γ+σ; 1− z1 − z2), (9)

where z1 > 0 and z2 > 0.

Note that the Gauss hypergeometric functions in the densities (6), (8) and (9) can be

expanded in series form if 0< z1+z2 < 1. However, if z1+z2 > 1, then 1−1/(z1+z2)< 1 and

we use (A.6) to rewrite the densities (6), (8) and (9) in series involving Gauss hypergeometric

functions having 1− 1/(z1 + z2) as argument.

The next theorem gives the density of the product of Kummer-beta and inverted hyperge-

ometric function type I variables.

Theorem 4. Let (X1, X2) and X3 be independent, (X1, X2)∼ IH I (ν1,ν2;α,β ,γ) and

X3 ∼ KB(κ,µ,λ). Then, the p.d.f. of (Z1, Z2) = (X1, X2)X3 is

Γ(ν + γ−α)Γ(ν + γ− β)Γ(µ+ κ)Γ(γ+ κ)

Γ(ν1)Γ(ν2)Γ(κ)Γ(γ)Γ(ν + γ−α−β)Γ(γ+µ+κ)
{1F1(µ;κ+µ;λ)}−1

×
z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

∞
∑

r=0

(α)r(β)r(γ+ κ)r

(γ+µ+ κ)r (γ)r r!
(1+ z1 + z2)

−r

×Φ1

�

µ,ν + γ+ r;γ+µ+ κ+ r;
1

1+ z1 + z2

,λ

�

, z1 > 0, z2 > 0.



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 324

Proof. The joint p.d.f. of (X1, X2) and X3 is given by

K2

x
ν1−1

1 x
ν2−1

2 xκ−1
3 (1− x3)

µ−1

(1+ x1+ x2)
ν+γ 2F1

�

α,β ;γ;
1

1+ x1+ x2

�

exp[λ(1− x3)], (10)

where x1 > 0, x2 > 0, 0< x2 < 1 and

K2 =
Γ(ν + γ−α)Γ(ν + γ− β)

Γ(ν1)Γ(ν2)Γ(γ)Γ(ν + γ−α− β)
{B(κ,µ)1F1(µ;κ+µ;λ)}−1.

Transforming Z1 = X1X3, Z2 = X1X3 and W = 1− X3 with the Jacobian

J(x1, x2, x3 → z1, z2, w) = 1/(1− w)2 in (10) and integrating w, we obtain the joint p.d.f. of

Z1 and Z2 as

K2

z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

∫ 1

0

wµ−1(1−w)γ+κ−1

�

1−w/(1+ z1 + z2)
�ν+γ

× exp(λw)2F1

�

α,β ;γ;
(1+ z1 + z2)

−1(1−w)

1−w/(1+ z1 + z2)

�

dw. (11)

Now, expanding Gauss hypergeometric functions in the integral (11) in terms of power series

we arrive at

K2

z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

∞
∑

r=0

(α)r(β)r

(γ)r r!
(1+ z1 + z2)

−r

∫ 1

0

wµ−1(1−w)γ+κ+r−1 exp(λw)
�

1−w/(1+ z1 + z2)
�ν+γ+r

dw.

Finally, applying (A.12) and substituting for K2 we obtain the desired result.

Corollary 4. Let (X1, X2) and X3 be independent, (X1, X2)∼ DII(ν1,ν2;γ) and X3 ∼ KB(κ,µ,λ).

Then, the p.d.f. of (Z1, Z2) = (X1, X2)X3 is given by

Γ(γ+ ν)Γ(µ+ κ)Γ(γ+ κ)

Γ(ν1)Γ(ν2)Γ(κ)Γ(γ)Γ(γ+µ+ κ)
{1F1(µ;κ+µ;λ)}−1

×
z
ν1−1

1
z
ν2−1

2

(1+ z1 + z2)
ν+γ
Φ1

�

µ,ν + γ;γ+µ+ κ;
1

1+ z1 + z2

,λ

�

, z1 > 0, z2 > 0.

Corollary 5. Let (X1, X2) and X3 be independent, (X1, X2) ∼ DI I(ν1,ν2;γ) and X3 ∼ BI (κ,µ).

Then, the p.d.f. of (Z1, Z2) = (X1, X2)X3 is given by

Γ(γ+ ν)Γ(µ+κ)Γ(γ+κ)

Γ(ν1)Γ(ν2)Γ(κ)Γ(γ)Γ(γ+µ+ κ)

z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

×2F1

�

µ,ν + γ;γ+µ+ κ;
1

1+ z1 + z2

�

, z1 > 0, z2 > 0.



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 325

Corollary 6. Let (X1, X2) and X3 be independent, (X1, X2)∼ IH I (ν1,ν2;α,β ,γ) and

X3 ∼ BI (κ,µ). Then, the p.d.f. of (Z1, Z2) = (X1, X2)X3 is given by

Γ(ν + γ−α)Γ(ν + γ− β)Γ(µ+ κ)Γ(γ+ κ)

Γ(ν1)Γ(ν2)Γ(κ)Γ(γ)Γ(ν + γ−α−β)Γ(γ+µ+ κ)

×
z
ν1−1

1
z
ν2−1

2

(1+ z1 + z2)
ν+γ

∞
∑

r=0

(α)r(β)r(γ+ κ)r

(γ+µ+ κ)r (γ)r r!
(1+ z1 + z2)

−r

×2F1

�

µ,ν + γ+ r;γ+µ+ κ+ r;
1

1+ z1 + z2

�

, z1 > 0, z2 > 0.

Theorem 5. Let (X1, X2) and X3 be independent, (X1, X2)∼ IH I (ν1,ν2;α,β ,γ) and

X3 ∼ BI I I(κ,µ). Then, the p.d.f. of (Z1, Z2) = (X1, X2)X3 is given by

Γ(ν + γ−α)Γ(ν + γ− β)Γ(κ+µ)Γ(κ+ γ)

2µΓ(ν1)Γ(ν2)Γ(κ)Γ(γ)Γ(ν + γ−α−β)Γ(κ+µ+ γ)

×
z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

∞
∑

r=0

(α)r(β)r(γ+ κ)r

(γ)r(γ+ κ+µ)r r!
(1+ z1 + z2)

−r

×F1

�

µ;ν + γ+ r,µ+ κ;γ+ κ+µ+ r;
1

1+ z1 + z2

,
1

2

�

,

where z1 > 0 and z2 > 0.

Proof. The joint p.d.f. of (X1, X2) and X3 is given by

K3

x
ν1−1

1 x
ν2−1

2 xκ−1
3 (1− x3)

µ−1

(1+ x1+ x2)
ν+γ(1+ x3)

κ+µ 2F1

�

α,β ;γ;
1

1+ x1+ x2

�

, (12)

where x1 > 0, x2 > 0, 0< x3 < 1 and

K3 =
Γ(ν + γ−α)Γ(ν + γ− β)

Γ(ν1)Γ(ν2)Γ(γ)Γ(ν + γ−α− β)
2κ{B(κ,µ)}−1.

Now, transforming Z1 = X1X3, Z2 = X2X3 and W = 1− X3 with the Jacobian

J(x1, x2, x3→ z1, z2, w) = 1/(1−w)2 in (12) and integrating w, the marginal p.d.f. of (Z1, Z2)

is derived as

K3 z
ν1−1

1 z
ν2−1

2

2κ+µ(1+ z1 + z2)
ν+γ

∫ 1

0

wµ−1(1−w)κ+γ−1

�

1−w/(1+ z1 + z2)
�ν+γ

(1−w/2)κ+µ

× 2F1

�

α,β ;γ;
(1+ z1 + z2)

−1(1−w)

1−w/(1+ z1 + z2)

�

dw. (13)

Expanding Gauss hypergeometric functions in the integral (13) in series form we arrive at

K3z
ν1−1

1
z
ν2−1

2

2κ+µ(1+ z1 + z2)
ν+γ

∞
∑

r=0

(α)r(β)r

(γ)r r!(1+ z1 + z2)
r

∫ 1

0

wµ−1(1−w)κ+γ+r−1

�

1−w/(1+ z1 + z2)
�ν+γ+r

(1−w/2)κ+µ
dw.

Finally, the desired result follows by using (A.11) and substituting for K3.



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 326

Corollary 7. Let (X1, X2) and X3 be independent, (X1, X2)∼ DI I(ν1,ν2;γ) and

X3 ∼ BI I I(κ,µ). Then, the p.d.f. of (Z1, Z2) = (X1, X2)X3 is given by

Γ(ν + γ)Γ(κ+µ)Γ(κ+ γ)

2µΓ(ν1)Γ(ν2)Γ(κ)Γ(γ)Γ(κ+µ+ γ)

z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

×F1

�

µ;ν + γ,µ+ κ;γ+κ+µ;
1

1+ z1 + z2

,
1

2

�

,

where z1 > 0 and z2 > 0.

Theorem 6. Let (X1, X2) and X3 be independent, (X1, X2)∼ IH I (ν1,ν2;α,β ,γ) and

X3 ∼ H I (κ,µ,ρ,σ). Then, the p.d.f. of (Z1, Z2) = (X1, X2)X3 is given by

Γ(ν + γ−α)Γ(ν + γ− β)Γ(κ+ γ)Γ(σ+ κ−µ)Γ(σ+ κ−ρ)

Γ(ν1)Γ(ν2)Γ(κ)Γ(γ)Γ(ν + γ−α− β)Γ(σ+ κ−µ−ρ)Γ(κ+ γ+σ)

×
z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

∞
∑

s=0

∞
∑

r=0

(µ)s(ρ)s(α)r(β)r(κ+ γ)r

(γ)r(κ+ γ+σ)s+r s! r!
(1+ z1 + z2)

−r

×2F1

�

σ+ s,ν + γ+ r;κ+σ+ γ+ s+ r;
1

1+ z1 + z2

�

,

where z1 > 0 and z2 > 0.

Proof. The joint p.d.f. of (X1, X2) and X3 is given by

K4 x
ν1−1

1 x
ν2−1

2 xκ−1
3 (1− x3)

σ−1

(1+ x1 + x2)
ν+γ 2F1

�

α,β ;γ;
1

1+ x1 + x2

�

2F1(µ,ρ;σ; 1− x3), (14)

where x1 > 0, x2 > 0, 0< x3 < 1 and

K4 =
Γ(ν + γ−α)Γ(ν + γ− β)

Γ(ν1)Γ(ν2)Γ(γ)Γ(ν + γ−α− β)

Γ(σ+ κ−µ)Γ(σ+ κ−ρ)

Γ(σ)Γ(κ)Γ(σ+ κ−µ−ρ)
.

Now, transforming Z1 = X1X3, Z2 = X2X3 and W = 1− X3 with the Jacobian

J(x1, x2, x3 → z1, z2, w) = 1/(1 − w)2 in (14) and integrating w, we obtain the p.d.f. of

(Z1, Z2) as

K4 z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

∫ 1

0

wσ−1(1−w)κ+γ−1

�

1−w/(1+ z1 + z2)
�ν+γ

×2F1

�

α,β ;γ;
(1+ z1 + z2)

−1(1−w)

1−w/(1+ z1 + z2)

�

2F1

�

µ,ρ;σ; w
�

dw, z1 > 0, z2 > 0.

Now, expanding the Gauss hypergeometric functions in series form, integrating the resulting

expression using (A.5), substituting for K4 and simplifying, we obtain the desired result.



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 327

Corollary 8. Let (X1, X2) and X3 be independent, (X1, X2)∼ DI I(ν1,ν2;γ) and X3 ∼ H I (κ,µ,ρ,σ).

Then, the p.d.f. of (Z1, Z2) = (X1, X2)X3 is given by

Γ(ν + γ)Γ(κ+ γ)Γ(σ+κ−µ)Γ(σ+ κ−ρ)

Γ(ν1)Γ(ν2)Γ(κ)Γ(γ)Γ(σ+ κ−µ−ρ)Γ(κ+ γ+σ)

z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

×
∞
∑

s=0

(µ)s(ρ)s

(κ+ γ+σ)s s!
2F1

�

σ+ s,ν + γ;κ+σ+ γ+ s;
1

1+ z1 + z2

�

,

where z1 > 0 and z2 > 0.

Theorem 7. Let (X1, X2), Y1 and Y2 be independent, (X1, X2)∼ IH I (ν1,ν2;α,β ,γ) and

Yi ∼ BI (ai, bi), i = 1,2. Then, the p.d.f of (Z1, Z2) = (X1, X2)Y1Y2 is given by

Γ(a1 + b1)Γ(a2+ b2)Γ(a1+ γ)

Γ(a1)Γ(a2)Γ(a1+ b1 + b2+ γ)

Γ(γ+ ν −α)Γ(γ+ ν − β)

Γ(ν1)Γ(ν2)Γ(γ)Γ(γ+ ν −α− β)

×
z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

∞
∑

r=0

∞
∑

s=0

(b2)s(a1+ b1 − a2)s(α)r(β)r(a1 + γ)r

(γ)r(a1 + b1 + b2 + γ)s+r s! r!
(1+ z1 + z2)

−r

×2F1

�

b1 + b2 + s,ν + γ+ r; a1+ b1 + b2+ γ+ s+ r;
1

1+ z1 + z2

�

,

where z1 > 0 and z2 > 0.

Proof. Using Theorem A.8, Y1Y2 ∼ H I (a1, b2, a1 + b1 − a2, b1 + b2). Now, using indepen-

dence of (X1, X2) and X3 and Theorem 6, we obtain the desired result.

Corollary 9. Let (X1, X2), Y1 and Y2 be independent, (X1, X2)∼ DI I(ν1,ν2;γ) and Yi ∼ BI (ai, bi),

i = 1,2. Then, the p.d.f of (Z1, Z2) = (X1, X2)Y1Y2 is given by

Γ(a1 + b1)Γ(a2 + b2)Γ(a1+ γ)Γ(ν + γ)

Γ(a1)Γ(a2)Γ(ν1)Γ(ν2)Γ(γ)Γ(a1+ b1 + b2+ γ)

z
ν1−1

1 z
ν2−1

2

(1+ z1 + z2)
ν+γ

×
∞
∑

s=0

(b2)s(a1 + b1− a2)s

(a1 + b1 + b2 + γ)s s!
2F1

�

b1 + b2 + s,ν + γ; a1 + b1 + b2+ γ+ s;
1

1+ z1 + z2

�

,

where z1 > 0 and z2 > 0.

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+ 1) · · · (a+ n− 1) = (a)n−1(a+ n− 1) for n= 1,2, . . . , (a)0 = 1.



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 328

The generalized hypergeometric function of scalar argument is defined by

pFq(a1, . . . , ap; b1, . . . , bq; z) =

∞
∑

k=0

(a1)k · · · (ap)k

(b1)k · · · (bq)k

zk

k!
, (A.1)

where ai, i = 1, . . . , p; b j, j = 1, . . . ,q are complex numbers with suitable restrictions and z is

a complex variable. Conditions for the convergence of the series in (A.1) are available in the

literature, see Luke [5]. From (A.1) it is easy to see that

0F0(z) =

∞
∑

k=0

zk

k!
= exp(z), 1F1(a; c; z) =

∞
∑

k=0

(a)k

(c)k

zk

k!
,

and

2F1(a, b; c; z) =

∞
∑

k=0

(a)k(b)k

(c)k

zk

k!
, |z| < 1.

Also, under suitable conditions, we have from Luke [5, Eq. 3.6(10)],

∫ 1

0

zα−1(1− z)β−1
pFq(a1, . . . , ap; b1, . . . , bq; z y)dz

=
Γ(α)Γ(β)

Γ(α+ β)
p+1Fq+1(a1, . . . , ap,α; b1, . . . , bq,α+ β ; y) (A.2)

and Luke [5, Eq. 3.6(13)],

∫ ∞

0

exp(−δz)zα−1
pFq(a1, . . . , ap; b1, . . . , bq; z y)dz

= Γ(α)δ−αp+1Fq(a1, . . . , ap,α; b1, . . . , bq;δ−1 y). (A.3)

The integral representations of the confluent hypergeometric function and the Gauss hyper-

geometric function are given as

1F1(a; c; z) =
Γ(c)

Γ(a)Γ(c − a)

∫ 1

0

ta−1(1− t)c−a−1 exp(zt)dt, (A.4)

and

2F1(a, b; c; z) =
Γ(c)

Γ(a)Γ(c− a)

∫ 1

0

ta−1(1− t)c−a−1(1− zt)−b dt, (A.5)

respectively, where Re(a)> 0 and Re(c − a)> 0. It is easy to check by using (A.5) that

2F1(a, b; c; z) = (1− z)−a
2F1

�

a, c − b; c;
−z

1− z

�

= (1− z)−b
2F1

�

c − a, b; c;
−z

1− z

�

(A.6)



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 329

and for Re(a) > 0, Re(b+ 1− c) > 0 and |arg(z)| < π, it has been shown that [see Luke 5,

Eq. 3.6.3],

2F1(a, b; a+ b+ 1− c; 1− z) =
Γ(a+ b+ 1− c)

Γ(b)Γ(a+ 1− c)

∫ ∞

0

sb−1(1+ s)c−b−1 ds

(1+ sz)a
. (A.7)

Further, for Re(λ)< Re(ν), we have [Prudnikov 12, Eq. 1.2.4.4],
∫ ∞

x

yλ−1

(y + a)ν
dy =

xλ−ν

ν −λ 2F1

�

ν ,ν −λ; 1+ ν −λ;−
a

x

�

. (A.8)

The Appell’s first hypergeometric function F1 is defined by

F1(a; b1, b2; c; z1, z2) =

∞
∑

r,s=0

(a)r+s(b1)r(b2)s

(c)r+s

zr
1zs

2

r! s!

=

∞
∑

r=0

(a)r(b1)r

(c)r

zr
1

r!
2F1(a+ r, b2; c + r; z2)

=

∞
∑

s=0

(a)s(b2)s

(c)s

zs
2

s!
2F1(a+ s, b1; c + s; z1), (A.9)

where |z1| < 1 and |z2| < 1. The Humbert’s confluent hypergeometric function Φ1 is defined

by

Φ1[a, b1; c; z1, z2] =

∞
∑

r,s=0

(a)r+s(b1)r

(c)r+s

zr
1zs

2

r! s!
,

=

∞
∑

r=0

(a)r(b1)r

(c)r

zr
1

r!
1F1(a+ r; c + r; z2)

=

∞
∑

s=0

(a)s

(c)s

zs
2

s!
2F1(a+ s, b1; c + s; z1), (A.10)

where |z1| < 1, |z2| <∞. The integral representations of F1 and Φ1 are given by

F1(a; b1, b2; c; z1, z2) =
Γ(c)

Γ(a)Γ(c − a)

∫ 1

0

va−1(1− v)c−a−1 dv

(1− vz1)
b1(1− vz2)

b2
, (A.11)

and

Φ1[a, b1; c; z1, z2] =
Γ(c)

Γ(a)Γ(c − a)

∫ 1

0

va−1(1− v)c−a−1 exp(vz2)dv

(1− vz1)
b1

, (A.12)

where Re(a) > 0 and Re(c − a) > 0. Note that for b1 = 0, F1 and Φ1 reduce to 2F1 and

1F1 functions, respectively. For properties and further results on these functions the reader is

referred to Luke [5] and Srivastava and Karlsson [15].

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 [1], Johnson, Kotz

and Balakrishnan [4], Nagar and Zarrazola [11], and Sánchez and Nagar [13].



Paula Bran-Cardona, Edwin Zarrazola and Daya Nagar / Eur. J. Pure Appl. Math, 5 (2012), 317-332 330

Definition A.1. The random variable X is said to have a beta type I distribution with parameters

(a, b), a > 0, b > 0, denoted as X ∼ BI (a, b), if its p.d.f. is given by

{B(a, b)}−1 x a−1(1− x)b−1, 0< x < 1,

where B(a, b) is the beta function given by

B(a, b) = Γ(a)Γ(b){Γ(a+ b)}−1.

Definition A.2. The random variable X is said to have a beta type II distribution with parameters

(a, b), denoted as X ∼ BI I (a, b), a > 0, b > 0, if its p.d.f. is given by

{B(a, b)}−1 x a−1(1+ x)−(a+b), x > 0.

Definition A.3. The random variable X is said to have a beta type III distribution with parame-

ters (a, b), denoted as X ∼ BI I I(a, b), a > 0, b > 0, if its p.d.f. is given by

2a{B(a, b)}−1 x a−1(1− x)b−1(1+ x)−(a+b), 0< x < 1.

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

xα−1(1− x)β−1 exp [λ(1− x)]

B(α,β)1F1(β ;α+β ;λ)
, 0< x < 1,

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 .

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 > 0, b > 0, c > 0, denoted as X ∼ DI(a, b; c), if their joint

p.d.f. is given by

{B(a, b, c)}−1 x a−1 y b−1(1− x − y)c−1, x > 0, y > 0, x + y < 1,

where B(a, b, c) is defined by

B(a, b, c) = Γ(a)Γ(b)Γ(c){Γ(a+ b+ c)}−1.

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 > 0, b > 0, c > 0, denoted as X ∼ DI I(a, b; c), if their joint

p.d.f. is given by

{B(a, b, c)}−1 x a−1 y b−1(1+ x + y)−(a+b+c), x > 0, y > 0.



REFERENCES 331

Definition A.7. The random variable X is said to have a hypergeometric function type I distri-

bution, 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−1(1− x)γ−1

2F1(α,β ;γ; 1− x), 0< x < 1,

where γ+ ν −α−β > 0, γ > 0 and ν > 0.

The following result (Gupta and Nagar [3], Nagar and Alvarez [8]) states that the hyper-

geometric function type I distribution can be obtained as the distribution of the product of

two independent beta type I variables.

Theorem A.8. Let X1 and X2 be independent, X i ∼ BI (ai, bi), i = 1,2. Then,

X1X2 ∼ H I (a1, b2, a1+ b1 − a2, b1 + b2).

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 [2], Gupta and Nagar [3], and Nagar and Gupta [10].

Acknowledgments The research work of DKN was supported by the Comité para el Desar-

rollo de la Investigación, Universidad de Antioquia research grant no. IN560CE.

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