the distribution of the linear combination of stochastic ...

V.B.L. Chaurasia et al. / International Journal of Engineering Science and Technology (IJEST)

THE DISTRIBUTION OF THE LINEAR COMBINATION OF STOCHASTIC VARIABLES CONCERNING TO CERTAIN TRANSCENDENTAL FUNCTIONS V.B.L. CHAURASIA1 and VINOD GILL2 1

Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India. E-mail: [email protected] 2 Department of Mathematics, Arya Institute of Engineering and Technology, Kukas, Jaipur-302028, Rajasthan, India. E-mail: [email protected] Abstract The object of this paper is to obtain a general distribution of the linear combinations of independent non-negative stochastic variables. This result generalizes a class of distribution of the linear combination of independent non-negative stochastic variables and further the result of Chaurasia and Kumar [4] as a particular case of the general distribution is discussed. It is also shown that the result of Mathai and Saxena [9] and the result of Srivastava [13] is also particular case of our result. The result obtained here is quite general in nature and is capable of yielding a large number of corresponding results merely by specializing the parameters involved therein. Key words: Distribution function, Moment generating function, H-function (Both Series representation and contour integral representation), Generalized hypergeometric function. 1. Introduction: The distribution of linear combinations of independent stochastic variables have been studied by various authors notably Amoroso [1], Robin [10], Stacy [14], Kabe [8], Mathai and Saxena [9] and Srivastava [13]. Here, we consider a linear combination

U  p1x1  p 2 x 2    p n x n

…(1.1)

of n independent non-negative Stochastic variables x1, x2,…,xn with the density functions of the form

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d 1

fi x i  

for

xi i

e

Ci

i x i

H

mi' n i' pi' qi'

  y x i  i i 

0  x i   and f i  x i   0

(d(i) D(i)  ' 1,pi  f (i) F(i)  ' 1,qi

   H mi  n i  z x i  p i q i  i i  

   b(i) B(i) 1,q  i (a (i) A (i) 1,p

i

…(1.2)

for other values of xi, where i = 1,2,…,n and the constant Ci is given by

  f h'(i)  v i'   (i) (i)   f j  Fj F(i)    j1 h' qi'   f h'(i)  v i'   (i) (i)   1  f j  Fj F(i)   j mi' 1   h' mi'

di

Ci  i

mi'



 

h'1 v' 0 i

 1

vi'

v i' Fh'(i)

.

  f h'(i)  v i'    (i) '  (i) ' (i) (i)   1  d j  D j F(i)   fh' (i) vi  i  fh' (i)vi       F j1 Fh' h' h'    y   ' i i pi   f h'(i)  v i'   (i) (i)   d j  D j F(i)   jn i' 1   h'

.

  mi n i 1  z i H p 1q i i  (  i  i 

n i'

(i) '   1d   f h'  vi     a (i) A (i)  i 1,pi  i i (i) Fh'    b(i) B(i) 1,q i

It is assumed that the parameters of (1.2) are such that

     

…(1.3)

f i  x i   0

The H-function introduced by Fox ([7], p.408) is defined as

  a,A)1,p m,n H p,q z (b,B) 1,q  

 1  i    z  d ,    2i i

…(1.4)

where m

 j1 q

   



jm 1

n

 b j  B j   1  a j  A j  j1

1  b j  B j 

p



j n 1

.

…(1.5)

 a j  A j  

for convergence conditions and other details, see Fox ([7], p.408). The series representation of H-function ([12], 1-6) defined as

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  e P E P   M M, N  H P,Q  x (f F     Q Q  g 1 G 0

 

 1 G  G  x

G



G ! Fg

…(1.6)

where M

N

j1 j g

j1

 G     f j  FjG   1  e j  E jG 

P   Q 1 f F e E             j j G  j j G   jM 1 j N 1

.

1

.

…(1.7)

and

G   f g  G)/Fg

…(1.8)

also N

P

M

Q

T   E i   E i   Fi   Fi N 1

1

1

…(1.9)

M 1

For other details of series representation of H-function, see ([12], 1-6). 2. A General Distribution of Linear Combination Theorem.

Let xi (i = 1,2,…,n) be n independent non-negative Stochastic variables with density function

given by (1.2). If

U  1 x1  2 x 2     n x n

denotes the linear combination, then the density function of U

is given by

n

h(u)   i 1

    1 C  i   

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  (i) (i) 1           '  vi jh' jh'  (i) ' (i) f v F     1     j1 j1  y i  h' i h'  ' ' (i) ' pi qi v i Fh'  (i) (i)  1   jh'    jh'    j n i' 1 j mi' 1  mi'

mi'



 

h'1 v' 0 i

n i'

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   n   i 1   

.

 m  i  h 1  

mi





vi  0

 j1 qi

ni



 (i)  jh

1   (i)  jh

j1

pi (i) 1   jh    (i)  jh jmi 1 jn i 1





(i)  h  (i)  h i (i) Bh

 n (i)    h 1  n i 1    u u     u n   2   (1)   (n)    (i)  1    n h h h 1 n    i 1   (i)      h      i 1 

zi 

(i) (i)  b h  vi   Bh

,

  v  1 i   vi    

       

…(2.1)

where

 (i) jh'

 d (i) j

 D (i) j

 (i) jh'

 f j(i)

 Fj(i)

 (i) jh

 a (i) j

 A (i) j

 (i) jh

 b (i) j

 B (i) j

 f h'(i)  v i'  Fh'(i)

 f h'(i)  v i'  Fh'(i)

,

 b (i)  vi  h B (i) h  b (i)  vi  h B (i) h



,

,

 (i)  d i   i  f h'(i)  v i'   Fh'(i)   i  b (i)  v i   B (i) h h h

,

and 2 is a hypergeometric series of n variables. For definition see Erdélyi et al. ([5], p.385). The result (2.1) is valid under the following conditions

(i)

 i  0 R( i   0 R

(d i   i  f h'(i)



v i'   Fh'(i)    i

 b (i)  j min R  (i)   0 j B   j 

for i = 1,…,n and j = 1,…,k. (ii)

B (i)  b (i)     B (i)  b (i)  v h j j h

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where i = 1,…,n; ,v = 0,1,… and j = h=1,…,k; j  h.

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

(iii)

j1

B (i) j

pi

zi

j1

i

  A (i)  0 for 0   j

pi

(i) qi (i)  A j A j  j1 j1

 



B (i)  j

(i) Bj

|   where i  1,2,..., n ;

and is positive for

zi i

 0

Proof. A moment generating function i xi can be obtained as

M  x  t)   i i



e

i x i t

d 1

xi i

Ci

0

.

mi  n i

H p q

i i

   m' 1  i   C i  h'1    .y

.

  z x i  i i 



vi' 0

 i

 a (i) A(i) 1,p  b(i) B(i) 1,q

i

i

H

mi' n i' pi' qi'

  dx  i 

  y x i  i i 

(d(i) D(i)  ' 1,pi  f (i) F(i)  ' 1,qi

   

Here t being real parameter

    vi'   1  j1 j1 ' ' qi pi v i' Fh'(i)    1   (i)jh'    (i)jh'   j mi' 1 j n i' 1  mi'



e

i x i

 (i)  jh'

(i) (i)  f h'  vi'   Fh'

n i'

1   (i)  jh'

 i  i t)

  zi mi n i 1  H p 1q i i  (   t  i i  i 

(i) (i)  di i  f h'  vi'   Fh' 

(i) '   1d   f h'  vi     a (i) A (i)  i 1,pi  i i (i) Fh'    b(i) B(i) 1,q i

     

…(2.3)

Since the stochastic variables are linearly independent, a moment generating function (t) for U is obtained as n

 t)   M  x  t) i i

i 1

mi'

n

 i 1

1 Ci

mi'



 

h'1 v' 0 i

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 j1 qi'

n i'

 (i)  1   (i)  jh'  jh' j1

pi' (i) 1   jh'    (i)  jh' ' ' j mi 1 j n i 1





 1

vi'

v i' Fh'(i)

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.y

.

 i

(i) (i)  f h'  vi'   Fh'

 i  i t)

  zi mi n i 1  H p 1q i i  (   t  i i  i 

(i) (i)  di  i  f h'  vi'   Fh' 

(i) '   1d   f h'  vi     a (i) A (i)  1,pi i  i i (i) Fh'    b(i) B(i) 1,q i

  .   

…(2.4)

Now on using the series expansion for the H-function ([12], 1-6) in (2.4) and after a little simplification, we get

n

 t)   i 1

.y

n

 i 1

 m  i  h 1  

   m' 1  i  C i  h'1     i

mi'



vi' 0

mi



vi  0

j1 qi'



 (i)  1   (i)  jh'  jh' 1   (i)  jh'

 j1 qi

  i  i t) ni



 (i)  jh

 1

j1

j mi' 1

(i) (i)  f h'  vi'   Fh'







n i'

pi'



j n i' 1

pi (i) 1   jh    (i)  jh j mi 1 jn i 1



v i' Fh'(i)

(i) (i)  di  i  f h'  vi'   Fh' 

1   (i)  jh

j1

  (i)  jh'

vi'

       



vi  (i) h  1 B (i) v i  h

 zi  i   ( i  i t)

   

(i) b h  vi (i) Bh

       …(2.5)

where

 (i)   (i)   (i)   (i) and  (i) jh' jh' jh jh h

are defined in earlier equations.

Now collecting the power of t in (2.5), we obtain the quantity n

n

 i 1

  i  i

(i)  t) h



 i 1

(i)  h

n (i)   n (i)   h n   h  t i1    i 1  (i)   i 1 h 

i

 n    i1 





  i  i  1     t  

(i)  h

…(2.6)



of which the inverse Laplace transform is ([6], p.238)

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n

 i 1

(i)  h

i

 n (i)     h   i 1   

n



u i1

(i)  h 1

n   1u  n u  (n) (i)  2  (1)        h h  h   i 1  1 n  

…(2.7)

Here 2 is a hypergeometric series of n variables. Since (t) is a m.g.f. of U, the density function h(u) of U is obtained by taking the inverse Laplace transform of (t). On using (2.7), the expression of h(u) is obtained without any difficulty. 3. Applications: As an application of the result (2.1), we derive its three interesting special cases (I)

Reducing the first H-function to unity in the main theorem, we get the result by Srivastava ([13], p.64).

(II)

Reducing the first H-function to unity and taking A(i) and B(i) = 1 in the main theorem, replacing i by

ia, i by i, di by i from i = 1,…,n and using the result

n    2 1   n    i   au,...,au   e au   i 1

…(3.1) in the main theorem, we obtain the result due to Mathai and Saxena ([9], p.163). The probability density function of a linear combination of generalized gamma variates and the results obtained by Amoroso [1], Robins [10] and Stacy [14] also follow as particular cases. (iii)

The result recently obtained by Chaurasia and Kumar [4] for i = 1 can be derived from our result on

giving suitable values to the parameters involved in the H-function (1.2). (iv)

Taking

m i'  1 n i' 1 p i'  1 q i'  2  i  1 and replacing x i by  x i

in

(1.2),

we

get

another known result obtained by Chaurasia and Kumar [4]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

L. Amoroso, Ricerchi Interno Alla Curva Die Redditi., Ann. Mat. Pura Appl. Ser.4, 21 (1925), pp.123-159. B.L.J. Braaksma, Asymptotic Expansions and Analytic Continuation for Barnes Integral, Compsitio Math., 15 (1964), pp.239-341. P.C. Consul, Exact Distributions of Likelihood Ratio Criteria, Ann. Math. Statist., 38 (1967), pp.1160-1169. V.B.L. Chaurasia and D. Kumar, Distribution of the Linear Combination of Stochastic Variables Pertaining to Special Functions, International Journal of Engineering Science and Technology, 2(4), 2010, 394-399. Erdélyi et al., Higher Transcendental Functions, Vol.1, McGraw-Hill Book Co. Inc., New York, 1953. Erdélyi et al., Tables of Integral Transform, Vol.1, McGraw-Hill Book Co. Inc., New York, 1954. Fox, The G and H-function as Symmetrical Fourier kernels, Trans. Amer. Math. Soc., 98 (1961), pp.395-429. D.G. Kabe, On the Exact Distributions of Likelihood Ratio Criteria, Ann. Math. Statist., 38 (1967), pp.1160-1169. A.M. Mathai and R.K. Saxena, On the Linear Combination of Stochastic Variables, Metrika, 20, Fasc. 3 (1973), pp.160-169. H. Robin, The Distribution of a Definite Quadratic Form, Ann. Math. Statist., 19 (1948), pp.266-270. M. Sharma, Fractional Integration and Fractional Differentiation of the M-Series. Fractional Calculus and Applied Analysis, 11 (2008), pp.187-191. P. Skibiński, Some Expansion Theorems for the H-function, Indian J. Math. 14 (1972), pp.1-6. T.N. Srivastava, On the Linear Combination of Stochastic Variables, Indian J. Pure Appl. Math., 14(1) (1983), pp.64-69. E.W. Stacy, A Generalisation of Gamma Distribution, Ann. Math. Statist., 33 (1962), pp.1187-1199.

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