______________
E-mail address: [email protected] Received July 16, 2019
726
Available online at http://scik.org
J. Math. Comput. Sci. 9 (2019), No. 6, 726-738
https://doi.org/10.28919/jmcs/4227
ISSN: 1927-5307
ON THE APPLICATIONS OF ALEPH ( ) -FUNCTION IN A SLIGHTLY
DIFFERENT TYPE-2 BETA DENSITY MODEL
YASHWANT SINGH
Government College, Kaladera, Jaipur, Rajasthan, India
Copyright © 2019 the author(s).This is an open access article distributed under the Creative Commons Attribution License,which permits
unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract. In the present paper, the author has studied about the structures which are the products and ratios of statistically independently distributed positive real scalar random variables.The author has derived the exact density of a slightly different Type-2 beta density by the Mellin Transform and Hankel Transform of the unknown density and afterthat the unknown density has been derived in terms of Aleph- functions by taking the inverse Mellin transform and Inverse Hankel Transform .A more general structure of Type-2 beta density has also been discussed. Some special cases in terms of I -function are also given.
Key words: type-2 Beta density; aleph-Function; I -function; H -function; Mellin transform; inverse Mellin transform; Hankel transform; inverse Hankel transform.
2010 Mathematics Subject Classification: 33C10, 44A15.
1.INTRODUCTION
The I-function introduced by Saxena [4] will be represented and defined in slightly different
, , :
( ) ( )
i i
m n p q R
I z =I z = (( ) () (1, )) 1, 1, 1, , , , ,
, : , , ,
1
( ) 2
j j n ji jin
pi
i i j j m ji ji m qi
a a
m n s
p q R b b
L
I z s z ds
i
+ +
−
=
(1.1)Where i= −1, z0and z−s =exp[ sin | |− z +iarg ]z where | |z represents the natural logarithm of | |z and argzis not the principal value. Here
(
) (
)
(
)
(
)
1 1
1 1
1
1 ( )
1
m n
j j j j
j j
R q p
ji ji ji ji
j m j n
i
b B s a A s
s
b B s a A s
= =
= + = +
=
+ − − =
− − +
(1.2)For R=1, the I-function reduces to the H-function.
The - function introduced by Suland et.al. [6] defined and represented in the following form:
,, : :[ ]
i i i
m n p q r
z z
= , 1, 1,
, ; :
1, 1,
( , ) ,[ ( , )]
( , ) ,[ ( , )]
i
i i
i
j j n i ji ji n p m n
p qi r
j j m i ji ji m q
a a
z
b b
+ +
=
1
( ) 2
s
L
s z ds
=
(1.3)
where= −1;
1 1
1 1 1
( ) (1 )
( )
(1 ) (
i i
m n
j j j j
j j
q p
r
i ji ji ji ji
i j m j n
b s a s
s
b s a s
= =
= = + = +
− − +
=
− + −
(1.4)
We shall use the following notation:
(
)
(
)
*
1, 1,
, ,[ , ]
i
j j n i ji ji n p
A = a a + ,B* =
(
bj,j)
1,m,[i(
bji,ji)
]m+1,qi The Mellin transform of f x( )denoted by M f x s
( );
or F s( )is given by
10
( ); s ( )
M f x s x f x dx
−
=
(1.5)The Hankel transform of f x( )denoted by H
f x p( );
or F pv( )is given by
0
( ); ( ) ( )
v v
H f x p x J px f x dx
=
(1.6)A real scalar random variable x is said to have a real type-1 generalized beta distribution,if the
1 1 ( )
(1 ) ; 0 1, 0, 0 ( ) ( )
0,
( )
x x x
elsewhere
f x
− −
+ −
=
(1.7)
where the parameters and are real. The following discussion holds even when and are complex quantities. In this case, the conditions become Re( ) 0, Re( ) 0 where Re(.)means the real part of (.).
A real scalar random variable x is said to have a real type-2 generalized beta distribution,if the
density is of the following form ([1], p.121, eq. (4.8)):
1 ( ) ( )
(1 ) ; 0 , 0, 0 ( ) ( )
0,
( )
x x x
elsewhere
f x
− − +
+ +
=
(1.8)
Where the parameters and are real. The following discussion holds even when and are complex quantities. In this case, the conditions become Re( ) 0, Re( ) 0 where Re (.) means the real part of (.).
2.GENERAL STRUCTURES
A real scalar random variable x is said to have a real generalized type-2 generalized beta
distribution,if the density is of the following form:
( )
1 ( ) 2 1
(1 ) ( , ; , ; ; ) 0,
( )
tm
t
tm A
x Ax F a b c w px tm
elsewhere
f x
+
− − +
+ +
+
+
=
(2.1)
For 0 x , 0, 0,A0,a0,b0,c0,1+Ax 0.
Where the parameters and are real. The following discussion holds even when and are complex quantities. In this case, the conditions become Re( ) 0, Re( ) 0 where Re (.) means the real part of (.).
The hth-moment of x, when x has the density in (2.1), is given by
( )
h htm h
E x
tm A
+ +
=
+
For Re(+tm h+ )0, 0.
When and hare real, the moments can exist for some values of halso such that + h 0. The Mellin transform of f x( )is obtained from (2.2) as:
11
( ) s
tm s
M f x
tm A
−
+ + −
=
+
(2.3)
For Re(+tm s+ − 1) 0,s= +v 2r+ 2 0, 0.
The unknown density f x( )is obtained in terms of -function by taking the inverse Mellin transform of (2.3). That is
1
1,0
0,1: : 1 1 , ( )
iR tm
A
f x x
tm
−−− + −
=
+
(2.4)
The Hankel transform of f x( )is obtained from (2.2) as:
11 ( ) v( ) s
tm s
H f x J p
tm A
−
+ + −
=
+
(2.5)
For Re(+tm s+ − 1) 0,s= +v 2r+ 2 0, 0.
The unknown density f x( )is obtained in terms of -function by taking the inverse Hankel transform of (2.5). That is
1
1 1,0
0,1: : 1 1 ,
( ) ( )
i
v R tm
A
f x J p A x
tm
−−− + −
=
+
(2.6)
Consider a set of real scalar random variables x1,...,xk, mutually independently distributed, where xjhas the density in (2.1) with the parameters j, j,A jj; =1,...,kand consider the product
1 2... k
In the standard terminology in statistical literature, the th
h moment of u, when uhas the density
in (2.1), is given by
( )
j
j
j h
j
j
tm h
E u
tm A
+ +
=
+
(2.8)
For Re(j+tm h+ )0,j 0;j=1,...,k
Then the Mellin transform of g u( )of uis obtained from the property of the statistical independent and is given by
1 11
( ) s ... s
k
M g u =M x − M x − (2.9)
1 11 ( )
j
j
k
j s
j
j
j
tm s
M g u
tm A
− =
+ + −
=
+
(2.10)
For Re(j+tm s+ − 1) 0,j 0,s= +v 2r+ 2 0
The unknown density f x( )is obtained in terms of -function by taking the inverse Mellin transform of (2.10). That is
1
1 ,0
0, : : 1 1
1 1 ,
( ) ; 1,...,
j
j
i j
j j
k k
k
k R tm
j j
j
j
A
g u A u j k
tm
−−− + −
= =
= =
+
(2.11)
Then the Hankel transform of g u( )of uis obtained from the property of the statistical independent and is given by:
1 1
2 2
[ ( )] v( ) v( ) ... k v( k)
H g u =H x J px H x J px H x J px (2.12)
1 11
( ) ( )
j
j
k
j
v j s
j
j
tm s
H g u J p
tm A
− =
+ + −
=
+
(2.13)
The unknown density f x( )is obtained in terms of -function by taking the inverse Hankel transform of (2.13). That is
1
1 ,0
0, : : 1 1
1 1 ,
( ) ( ) ; 1,...,
j
j
i j
j j
k k
k
v j k R j tm
j
j
A
g u J p A u j k
tm
−−− + −
= =
= =
+
(2.14)
If we consider more general structures in the same category. For example, consider the structure
1 2
1 1 2 ... , 0, 1,...,
k
k j
u =x x x j= k (2.15) Where x1,...,xkare mutually independently distributed as in (2.1).
Then the Mellin transform of g u( )1 of u1 is given as
1( 1) ( 1)1 1
( ) s ... k s
k
M g u =M x − M x − (2.16)
1
1 ( 1)( 1) ( )
j j
j j
k
j s j
j
j
tm s
M g u
tm A
− =
+ + −
=
+
(2.17)
For Re(j+tm s+ − 1) 0,s= +v 2r+2,j 0.
The unknown density g u( )1 is obtained in terms of -function by taking the inverse Mellin transform of (2.17). That is
,0
1 0, : : 1
1 1 ,
( ) ; 1,...,
j
j j
j
i j j j
j j
k k
k
k R tm
j j
j
j
A
g u A u j k
tm
−−−
+ −
= =
= =
+
(2.18)
For Re(j+tm s+ − 1) 0,s= +v 2r+2,j 0.
The Hankel transform of g u( )1 of u1is obtained from the property of the statistical independent and is given by:
1 1 2 2
1 1 1 2 2
[ ( )] ( ) ( ) ... k ( k)
v v k v k
1
1 ( 1)( 1)
( ) ( )
j j
j j
k
j
v j s
j
j
tm s
H g u J p
tm A
− =
+ + −
=
+
(2.20)
For Re(j+tm s+ − 1) 0,s= +v 2r+ 2 0,j 0.
The unknown density g u( )1 is obtained in terms of -function by taking the inverse Hankel transform of (2.20). That is
,0
1 1 0, : : 1 1
,
( ) ( ) ; 1,...,
j
j j
j
i j j j
j j
k k
k
v j k R j tm
j
j
A
g u J p A u j k
tm
−−−
+ −
= =
= =
+
(2.21)
For Re(j+tm s+ − 1) 0,s= +v 2r+ 2 0,j 0.
A More General Structure
We can consider more general structures. Let
1 2 1 , ,...,
,...,
r
r k
x x x
w
x+ x
= (2.22)
Wherex1,...,xk, mutually independently distributed real random variables having the density in (2.1) with xjhaving parameters j, j,A jj; =1,...,k.
Then the Mellin transform of g w( )is given by
1 1 ( 1) ( 1)1 1
( ) s ... rs r s ... k s
M g w =M x − M x − M x − −+ M x − − (2.23)
1 1 11
1 ( )
j j
j
k r
j s
j j
j
j
tm s
M g w
tm
A A
−
= =
+ + −
=
+
1
( 1)
j
j
k
j s j r
tm s
A
− = +
+ − −
(2.24)
The unknown density g w( )is obtained in terms of -function by taking the inverse Mellin transform of (2.24). That is
1
1 ,0
0, : : 1 1
1 1 ,
( ) ; 1,...,
j
j
i j
j j
k r
r
r R tm
j j
j
j
A
g w A w j r
tm
−−− + −
= =
= =
+
1 1 1
1 ,
,0 0, : :
1 ; 1,...,
j
j j
j i
tm k
k r k r R
j r A w j r k
+ +
−
−
− = + −−−
= +
(2.25)
Then the Hankel transform of g w( ) is given as:
1 1
[ ( )] v( ) ... r v( r)
H g w =H x J px H x J px
H x Jr11 v
(
pxr11)
...H x Jk1 v( )
pxk1− − − −
+ +
(2.26)
1 1 11
1
( ) ( )
j j
j
k r
j
v j j s
j
j
tm s
H g w J p
tm
A A
−
= =
+ + −
=
+
1
( 1)
j
j
k
j s j r
tm s
A
− = +
+ − −
(2.27)
For Re(j+tm −(s 1))0,s= +v 2r+ 2 0
The unknown density g w( )is obtained in terms of -function by taking the inverse Hankel transform of (2.27). That is
1
1 ,0
0, : : 1 1
1 1 ,
( ) ( ) ; 1,...,
j
j
i j
j j
k r
r
v j r R j tm
j
j
A
g w J p A w j r
tm
−−− + −
= =
= =
+
1 1 1
1 ,
0, ,0: :
1 ; 1,...,
j
j j
j i
tm k
k r k r R
j r A w j r k
+ +
−
−
− = + −−−
= +
We can consider more general structures in the same category. For example, consider the
structure
1 1 1 1
1 ,...,
,...,
r
k r
r
r k
x x
w
x x
+
+
= (2.29)
wherex1,...,xk, mutually independently distributed real random variables having the density in (2.1) with xjhaving parameters j, j;j=1,...,k.
Then the Mellin transform of g w( 1)is given by
1( 1) ( 1) 1( 1) ( 1)1 1 1
( ) s ... r s r s ... k s
r r k
M g w M x − M x − M x−+ − M x− −
+
= (2.30)
1
1 1( 1)
1
( )
j j
j j
j j
k r
j s
j j
j
j
tm s
M g w
tm
A A
−
= =
+ + −
=
+
1
( 1)
j j
j j
k
j s j r
tm s
A
− = +
+ − −
(2.31)
For Re(j+tm −(s 1))0,s= +v 2r+ 2 0,j 0.
The unknown density g w( 1)is obtained in terms of -function by taking the inverse Mellin transform of (2.31). That is
,0
1 0, : : 1
1 1 ,
( ) ; 1,...,
j
j j
j
i j j j
j j
k r
r
r R tm
j j
j
j
A
g w A w j r
tm
−−−
+ −
= =
= =
+
1 ,
0,
,0: : 1 1 ; 1,...,
j j j
j
j j
j i
tm k
k r
k r R j r A w j r k
+ +
− −
−
− = + −−−
= +
(2.32)
For Re(j+tm −(s 1))0,s= +v 2r+ 2 0,j 0.
Then the Hankel transform of g w( 1) is given as:
1 1
1 1 1
[ ( )] ( ) ... r ( r)
v r v r
1
(
1)
(
)
1 1 ...
k k
r r
r v r k v k
H x−+ J px−+ H x− J px−
+ +
(2.33)
1
1 1
( 1)
1
( ) ( )
j j
j j
j j
k r
j
v j j s
j
j
tm s
H g w J p
tm
A A
−
= =
+ + −
=
+
1
( 1)
j j
j j
k
j s j r
tm s
A
− = +
+ − −
(2.34)
For Re(j+tm −(s 1))0,j 0,s= +v 2r+ 2 0
The unknown density g w( 1)is obtained in terms of -function by taking the inverse Hankel transform of (2.34). That is
,0
1 0, : : 1
1 1 ,
( ) ( ) ; 1,...,
j
j j
j
i j j j
j j
k r
r
v j r R j tm
j
j
A
g w J p A w j r
tm
−−−
+ −
= =
= =
+
1 ,
0,
,0: : 1
1 ; 1,...,
j j j
j
j j
j i
tm k
k r k r R
j r A w j r k
+ + −
−
−
− = + −−−
= +
(2.35)
For Re(j+tm −(s 1))0,j 0,s= +v 2r+ 2 0.
3.SPECIAL CASES
If we take i =1in (2.4), the unknown density f x( ) is obtained in terms of I -function. That is
1
1 1,0
0,1: 1 1
,
( ) A R tm
f x I A x
tm
−−− + −
=
+
(3.1)
For R=1, the I -function reduces to the H-function.
1
1 1,0
0,1: 1 1
,
( ) v( ) R tm
A
f x J p I A x
tm
−−− + −
=
+
(3.2)
For R=1, the I -function reduces to the H-function.
If we take i =1in (2.11), the unknown density g u( ) is obtained in terms of I -function. That is
1
1 ,0
0, : 1 1
1 1 ,
( ) ; 1,...,
j
j j
j j
k k
k
k R tm
j j
j
j
A
g u I A u j k
tm
−−− + −
= =
= =
+
(3.3)
For R=1, the I -function reduces to the H-function.
If we take i =1in (2.14), the unknown density g u( ) is obtained in terms of I -function. That is
1
1 ,0
0, : 1 1
1 1 ,
( ) ( ) ; 1,...,
j
j j
j j
k k
k
v k R tm
j j
j
j
A
g u J p I A u j k
tm
−−− + −
= =
= =
+
(3.4)
For R=1, the I -function reduces to the H-function.
If we take i =1in (2.18), the unknown density g u( )1 is obtained in terms of I -function. That is
,0
1 0, : 1
1 1 ,
( ) ; 1,...,
j
j j
j
j j j
j j
k k
k
k R tm
j j
j
j
A
g u I A u j k
tm
−−−
+ −
= =
= =
+
(3.5)
For R=1, the I -function reduces to the H-function.
If we take i =1in (2.21), the unknown density g u( )1 is obtained in terms of I -function. That is
,0
1 0, : 1
1 1 ,
( ) ( )
j
j j
j
j j j
j j
k k
k
v k R tm
j j
j
j
A
g u J p I A u
tm
−−−
+ −
= =
=
+
(3.6)
For R=1, the I -function reduces to the H-function.
1
1 ,0
0, : 1 1
1 1 ,
( ) ; 1,...,
j j j j j k r r
r R tm
j j
j
j
A
g w I A w j r
tm −−− + − = = = = + 1 1 1 1 , 0, ,0:
1 ; 1,...,
j j j j tm k k r k r R
j r
I A w j r k
+ + − − − = + −−− = + (3.7)
For R=1, the I -function reduces to the H-function.
If we take i =1in (2.28), the unknown density g w( ) is obtained in terms of I -function . That is
1
1 ,0
0, : 1 1
1 1 ,
( ) ( ) ; 1,...,
j j j j j k r r
v r R tm
j j
j
j
A
g w J p I A w j r
tm −−− + − = = = = + 1 1 1 1 , 0, ,0:
1 ; 1,...,
j j j j tm k k r k r R
j r
I A w j r k
+ + − − − = + −−− = + (3.8)
For R=1, the I -function reduces to the H-function.
If we take i =1in (2.32), the unknown density g w( 1) is obtained in terms of I -function. That is
,0
1 0, : 1
1 1 ,
( ) ; 1,...,
j
j j
j
j j j
j j
k r
r
r R tm
j j
j
j
A
g w I A w j r
tm −−− + − = = = = + 1 , 0, ,0: 1
1 ; 1,...,
j j j
j j j j tm k k r k r R
j r
I A w j r k
+ + − − − − = + −−− = + (3.9)
For R=1, the I -function reduces to the H-function.
If we take i =1in (2.35), the unknown density g w( 1) is obtained in terms of I -function. That is
,0
1 0, : 1
1 1 ,
( ) ( ) ; 1,...,
j
j j
j
j j j
j j
k r
r
v j r R j tm
j
j
A
g w J p I A w j r
0, ,0: 1
1 1 , ; 1,...,
j j
j j j
j j
k k r k r R
j r
I A w j r k
−
− −−−
− = + +
−
= +
(3.10)
For R=1, the I -function reduces to the H-function.
Conflict of Interests
The author(s) declare that there is no conflict of interests.
REFERENCES
[1] Mathai, A.M.; A pathway to matrix-variate gamma and normal densities, Linear Alg. Appl.396 (2005), 317-328.
[2] Mathai, A.M. and Saxena,R.K.; The H-function with Applications in Statistics and Other Disciplines, Wiley Eastern, New Delhi and Wiley Halsted, New York (1978).
[3] Mathai, A.M., Saxena, R.K. and Haubold, H.J.; The H-function: Theory and Applications, CRC Press, New York (2010).
[4] Saxena, V.P.; Formal solution of certain new pair of dual integral equations involving H-functions, Proc. Nat. Acad. India, Sect. A 52(1982), 366-375.
[5] Srivastava, H.M., Gupta, K.C. and Goyal, S.P; The H-function of One and Two Variables with Applications, South Asian Publisher, New Delhi, (1982).
