Applied Mathematics, 2011, 2, 843-845
doi:10.4236/am.2011.27113 Published Online July 2011 (http://www.SciRP.org/journal/am)
A New Bivariate Gamma Distribution
Arjun K. Gupta1, Dattatraya G. Kabe2
1Bowling Green State University, Bowling Green, USA
25971 Greensboro Drive , Mississauga , Canada
Received March 27, 201 1; revised May 18, 2011; accepted May 21, 2011
Abstract
Following Nadarajah [1], we introduce a new bivariate correlated type Gamma distribution, whose joint
density is expressed in two parts. Expressions for single and joint moments of the variates are derived.
Bivariate Correlated Wishart density follows on similar lines.
Keywords: Gamma Distribuion, Correlated Gamma Variates, Moments, Gauss’ Hypergeometric Series
1. Introduction
We have joint density of three independent gamma
random variables
  
11 1
,, =exp
ab c
huvwKuvwu v w
 


(1)
where , as a generic letter, denotes the normalizing
constant of a denstiy function in this paper. Setting
K
=,=uxwuyw the joint denstiy at ,,
x
yw is

  
11
1
,,
=exp
ab
c
huvw

.
K
xwywwxy w


 
(2)
As in Nadarajah [1], if <
x
y
=wy
, then we set ,
and if , then we set , and integrate out .
This is done by using Gr adshteyn and Ryzhik ([ 2], 3.385)
result , and we find that (See Equation (3))
=wxt
t
<yx t
where,




=0 =0
1
,,,, =!!
=2;; ;
j
k
jk k
jk jk
x
y
rxy rjk
Frxy



 


 (4)
However, Nadarajah [1] does not express the density
(3) in terms of 21
F
.
The series is one of the several of
Lauricella's series, and Mathai ([3], Theroem 5.59) gives

,,,,rxy 


an integral representaion of

. We use this integral
representation to show that

can be expressed as 1 2
F
.
To calculate single or joint moments of
x
and ,
we set y


1221 22
=,=;,:,=,if <
x
yJxy x   y (5)


1221 22
=,=;,:,=,if < .
y
xJxy x   y (6)
Similarly , if <
x
y , then we set =yxz

,
0< <
x
z and integrate out to obtain the marginal
denstiy of z
x
. However, if , then we must
integrate out from 0 to <yx y
x
, by using certain known
integral results like
 

1
0=0 0
!
expd =exp!
x
k
p
xp
pk
y
p
yy yyk





 




 (7)
to obtain the marginal density of
x
. However, all such
results are already available in advanced calculus books
or in the books of collected results on integrals and
series.
2. The Moments
Mathai [4] shows that the Lauricella’s series can be
written as
 

 
11
11
exp,1,,,if <
,=
exp,1,,,if < .
ac b
abc
x
K
xyxy cbacyxy
y
hxy y
K
xyxyc abcxyx
x





 



 



(3)

A. K. GUPTA ET AL. 844
   

1
11
11
111 1
0
111 1
1
=0 =0112
,, ,;;, ,=111dd
()()()
=.
!!!
n
b ca b
an
Dnn n
r r
nnn
n
n
rr nn
Fabbc xxKuuuxuxuu
arrb rbr
n
x
x
crrrr r
  


 
  



 



(8)
He ([3,4]) also records several integral representation
of the series
D
F
. The context one is Mathai ([3],
Theorem 5.59, p. 3 45 )

 

1
11
1
11
111 1
,,;;, ,
=exptr
;;d d
Dn n
bgbg n
nn
nn n
Fab bcXX
KTTTT
FacXTXT TT







(9)
where 11 ,,,,,
nn
X
XT T apos
pos definite sym-
m
re ppitive definite
symmetric matrices and

2= 1gp.
If A and B are two itive
pp
nonceetric matrices having ntral Wishart densities
with respective noncentrality parameter matrices

and

and degree of freedom n and q, then w
knowat e th










1
11
0 00
=exptr;; dd=exptr;,
ng qgnqg
F
ABD
A
BA BFnFqABKDDnqD
 
 
 (10)
We write some what incorrectly , but formally the re sult (10) as









0110 1
0
;;=; .
F
nAFqBFnqAB 
(11)
Now from (9)
 


 


  
0
0
0
0
11111 111
11 11011
112 121
11 11011
;;F acX=exp tr;ddd
=exp tr;;ddd
=exp tr(;()())ddd
=exp tr;;ddd
ag
nnnn n
ag
nnn n
ag
nnn
ag
nnn n
TXTZZFcXTXTZTT
ZZFcXTZFcXTZZTT

Z
ZFcXXXTT TZZTT
ZZFcXTZFcXTZZTT



 













11121 2
=;;nn FacXXXT TT 
(12)
where and hence from (9) and (12) we have
1
=n cc c



112211 12
;, ,;;,, ,=;;;.
Dn nnn
F
abbcXXXFabbcXXX  
(13)
The marginal density of , setting z
x
y

=
x
, if <
x
y , is
 





 

 
11
111
=0
1
1
=0
=exp2d
=exp
!
=exp2;;
!
ab
ar br c
r
r
hxKxwxw zx z wzw
br
d
2dd
K
xwz wxzwzw
r
br br
KxFcac
rrx


 

 
 
 



(14)
Similarly, usi ng (7) we can find the m argi nal density of
x
, when <
x
y. Then using (5),(6) and (14) we find that



 
21 1
3
21 1
3
1
<<=1,;1;,1;;
2
1
1,;1;, 1;;;1
2
mn mn
ExyxyExyy xFbnmncFcbmncbac ;1
F
anmncFc amncabc





 



(15)
The integral in (15) is evaluated by using first (5), (6), and then




11
11
2
011 d=1;1;;
1
nmnp mnp
m
xx xxKFnmn

 .

 

 



 (16)

A. K. GUPTA ET AL.845
Note that (15) consists of a double series and not a
triple series as in Nadarajah ([1], Section 3). Although he
1], Section 5) wishes to generalize bivariate correlated
a
[1] S. Nadarajah, “A New Bivariate Chi-Distribution,” SIAM
(unpublished).
[2] I. S. Gradsheleyn and I. M. Ry
Series, and Products,” 5th Editi
York, 2000.
([
gmma densities to n variate similar gamma densities,
it is not a simple task. It will involve complicated order
statistics theory from different Populations.
3. References
Review, 2006,
zhik, “Tables of Integral,
on, Academic Press, New
[3] A. M. Mathai, “Jacobian of Transformations and Func-
tions of Matrix Argument. World Scientific,” London,
1997.
[4] A. M. Mathai, “A Handbook of Generalized Special
Functions for Statistical and Physical Sciences,” Claredon
Press, Oxford, 1993.