SOOCHOW JOURNAL OF MATHEMATICS

Volume 31, No. 2, pp. 205-211, April 2005

ON THE CONVOLUTION OF GAMMA DISTRIBUTIONS BY MOHAMED AKKOUCHI Abstract. In this paper, we give a formula for the distribution of the sum of n independent random variables with gamma distributions. A formula for such a sum was provided by Mathai (see [5]) in 1982. But it was complicated. In the paper [1], Jasiulewicz and Kordecki derived a formula for the particular case of independent random variables having Erlang distributions by using Laplace transform. Our method is based on elementary computations and our result (see Theorem 2.1 below) is expressed by means of the generalized beta function.

1. Introduction The distribution of the sum of n independent exponentially distributed random variables with different parameters β i (i = 1, 2, . . . , n) is given in [2], [3], [4] and [6]. In the paper [1], H. Jasiulewicz and W. Kordecki have given the distributions of this sum without assuming that all the parameters β i are different. They reduced the problem to the one of finding the distribution of the sum of independent random variables having Erlang distributions. All of the above problems are the special cases of a sum of independent random variables with gamma distributions. In the paper [5], A.M. Mathai has provided a formula for such a sum, but as it was noticed in [1] and [6], this formula is complicated even in the case where the random variable are exponentially distributed. We point out that in the papers [1] and [6] the authors derived directly their results instead of using the results obtained in [5]. We point out also that the methods used by the authors in [1] to get their results are based on the use of Laplace transform. Received October 2, 2003. AMS Subject Classification. 60E05, 60G50. Key words. independent random variables, convolution, gamma distributions, generalized beta function. 205

206

MOHAMED AKKOUCHI

The aim of this note is a contribution to the investigations of the distribution of the sum of n independent random variables having gamma distributions. Our main result is presented in Section two (see Theorem 2.1 below). Our method is based on some elementary computations and our result is expressed by means of a multiple integral involving the generalized beta function. 2. The Result In all this note, X1 , . . . , Xn will be n independent random variables having gamma distributions. More precisely, we suppose that every X i has a probability density function (pdf) fXi given by fXi (t) :=

βi αi αi −1 t exp(−tβi ) χ[0,∞] (t), Γ(αi )

(1)

where Γ is the usual gamma function, χ [0,∞] (t) = 1 if t > 0 and χ[0,∞](t) = 0 elsewhere, and the numbers αi , βi are positive for all i = 1, 2, . . . , n. We say that Xi has a gamma distribution γ(αi , βi ) with parameters αi and βi for all i = 1, 2, . . . , n. We would like to find the distribution of the random variable: Sn := X1 + X2 + · · · + Xn .

(2)

We recall that for every complex number z with positive real part, Γ(z) is given by

Z

Γ(z) =

∞

tz−1 exp(−t) dt.

(3)

0

We recall that the beta function B(z 1 , z2 ) is given for all complex numbers z1 , z2 with positive real parts by B(z1 , z2 ) =

Z

1 0

tz1 −1 (1 − t)z2 −1 dt.

(4)

The following useful identities: B(z1 , z2 ) = 2

Z

π 2

cos2z1 −1 (t) sin2z2 −1 (t) dt,

(5)

0

and B(z1 , z2 ) =

Γ(z1 ) Γ(z2 ) = B(z2 , z1 ) Γ(z1 + z2 )

(6)

ON THE CONVOLUTION OF GAMMA DISTRIBUTIONS

207

are well known. For all integer n greater or equal to 2, we recall that the generalized beta function Bn is defined for all complex numbers z1 , z2 , . . . , zn , with positive real parts by Bn (z1 , z2 , . . . , zn ) =

Γ(z1 ) Γ(z2 ) · · · Γ(zn ) . Γ(z1 + z2 + · · · + zn )

(7)

The aim of this note is to prove the following result. Theorem 1. The probability distribution function f Sn of the random variable Sn defined by (2) is given by the formula fSn (t) = Ctα1 +···+αn −1 ×

Z

1 0

···

Z

1 0

e−tCβ1 ,...,βn (u1 ,...,un−1 ) Bα1 ,...,αn (u1 , . . . , un−1 )du1 · · · dun−1 , (8)

for all t > 0 and fSn (t) = 0 for all t ≤ 0, where C=

β1α1 β2α2 · · · βnαn , Γ(α1 + α2 + · · · + αn )

(9)

and Cβ1 ,...,βn (u1 , . . . , un−1 ) := β1

n−1 Y j=1

uj +

n−1 X i=2

βi (1 − ui )

n−1 Y j=i

uj + βn (1 − un−1 ), (10)

and Bα1 ,...,αn (u1 , . . . , un−1 ) :=

n−1 Y α1 +···+αj −1 1 (1 − uj )αj+1 −1 , (11) u Bn (α1 , . . . , αn ) j=1 j

for all u1 , . . . , un−1 ∈ [0, 1]. Proof. In order to prove the result above, we are led to compute the integrals In (g) := ×

Z

∞ 0

β1α1 · · · βnαn Γ(α1 ) · · · Γ(αn )

···

Z

∞

0

g(x1 + · · · + xn )xα1 1 −1 · · · xnαn −1 e−(β1 x1 +···+βn xn ) dx1 · · · dxn , (12)

for all g ∈ Cb (R) the space of all continuous and bounded functions on the real

line. To compute (12), we set xi := yi 2 where 0 ≤ yi < ∞ for all i = 1, 2, . . . , n.

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MOHAMED AKKOUCHI

With these new variables we have In (g) = Z

×

∞ 0

···

2n β1α1 · · · βnαn Γ(α1 ) · · · Γ(αn )

Z

∞

0

g(y1 2 +· · ·+yn 2 )y12α1 −1 · · · yn2αn −1 e−(β1 y1

2 +···+β

n yn

2)

dy1 · · · dyn .

(13)

To compute the integral (13), we shall use the spherical coordinates. Thus, we set y1 = r sin φn−1 · · · sin φ3 sin φ2 sin φ1 y2 = r sin φn−1 · · · sin φ3 sin φ2 cos φ1 y3 = r sin φn−1 · · · sin φ3 cos φ2 .. .. .. . . . yn−1 = r sin φn−1 cos φn−2 yn = r cos φn−1 , where 0 ≤ r < ∞, and 0 ≤ φk ≤

k = 1, 2, . . . , n − 1, by setting

rk2

π 2

:=

for all k = 1, . . . , n − 1. Conversely, for all

y12

+ · · · + yk2 and rn := r, we have

cos(φk ) =

yk+1 , rk+1

sin(φk ) =

rk . rk+1

and

We recall that the Jacobian of this change of variables is given by r n−1 sinn−2 φn−1 sinn−3 φn−2 · · · sin φ2 .

(14)

With these spherical coordinates, we have n Y

yi2αi −1 = r 2(α1 +···+αn )−n

i=1

n−1 Y

sin2(α1 +···+αj )−j cos2αj+1 −1 .

(15)

j=1

We observe also that β1 y1 2 + · · · + βn yn 2 = Cβ1 ,...,βn (sin2 (φ1 ), . . . , sin2 (φn−1 )),

(16)

ON THE CONVOLUTION OF GAMMA DISTRIBUTIONS

209

where the function Cβ1 ,...,βn is defined in (10). Taking into account the previous identities, we have 2n β1α1 · · · βnαn In (g)= Γ(α1 )· · ·Γ(αn ) ×

"n−1 Y

Z

"Z

∞ 2

g(r )r 0

2(α1 +···+αk )−1

sin

π 2

2(α1 +···+αn )−1

φk cos

2αk+1 −1

k=1

0

···

Z

π 2

e−r

2C

β1 ,...,βn (sin

2

φ1 ,...,sin2 φn−1 )

0

#

#

(17)

φk dφ1 · · · dφn−1 dr.

We set t = r 2 and uj = sin2 φj for all j = 1, . . . , n − 1. Then the Jacobian of this

mapping is given by

Y 1 1 n−1 √ . √ p n u 2 t j=1 j 1 − uj

Therefore β1α1 · · · βnαn In (g)= Γ(α1 )· · ·Γ(αn ) "n−1 Y

×

uk

Z

∞

g(t)t

α1 +···+αn−1

0

α1 +···+αk −1

k=1

(1 − uk )

Z

αk+1 −1

1 0

Z

(18)

1

· · · e−tCβ1 ,...,βn (u1 ,...,un−1 )

#

0

#

(19)

du1 · · · dun−1 dt.

Using the notations (11), the identity (19) gives β1α1 · · · βnαn In (g)= Γ(α1 +· · ·+αn ) ×

Z

1

0

Z

1

··· e

Z

∞

g(t)tα1 +···+αn −1 0

−tCβ1 ,...,βn (u1 ,...,un−1 )

0



Bα1 ,...,αn (u1 , . . . , un−1 ) du1 · · · dun−1 dt. (20)

(20) shows that the random variable S n has a probability density function f Sn given by fSn (t) = 0 if t ≤ 0 and fSn (t)= ×

Z

1 0

β1α1 · · · βnαn tα1 +···+αn −1 Γ(α1 + · · · + αn )

···

Z

1

0

e−tCβ1 ,...,βn (u1 ,...,un−1 ) Bα1 ,...,αn (u1 , . . . , un−1 )du1 · · · dun−1 ,

(21)

for all t > 0. Thus our theorem is completely proved. Remark 2. For all (α1 , . . . , αn ) ∈ Rn , we have Z

1 0

···

Z

1 0

Bα1 ,...,αn (u1 , . . . , un−1 ) du1 · · · dun−1 = 1.

(22)

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MOHAMED AKKOUCHI

Proof. For all (α1 , . . . , αn ) ∈ Rn , we have the following equalities Z

= = =

1 0

···

Z

n−1 YZ 1 k=1 0 n−1 Y k=1 n−1 Y

1 n−1 Y

0 k=1

uk α1 +···+αk −1 (1 − uk )αk+1 −1 du1 · · · dun−1

uk α1 +···+αk −1 (1 − uk )αk+1 −1 duk

B(α1 + · · · + αk , αk+1 )

Γ(α1 + · · · + αk )Γ(αk ) Γ(α1 ) · · · Γ(αn ) = . Γ(α + · · · + α ) Γ(α + · · · + α ) 1 1 n k+1 k=1

Taking into account (7) and (11), our remark is proved. We remark also that if all the βi are equal to a number β > 0, then for all u1 , . . . un ∈ [0, 1], we have Cβ,...,β (u1 , . . . , un−1 ) = β. With the remarks made above, we have the following corollary. Corollary 3. Suppose that all the βi are equal to a number β > 0, then the random variable Sn has a probability density function f Sn given by fSn (t) = 0 if t ≤ 0 and β α1 +···+αn tα1 +···+αn −1 e−βt , (23) fSn (t) = Γ(α1 + · · · + αn ) for all t > 0. Remark. (23) shows that under the assumptions of the previous corollary Sn , has a gamma distribution γ(α1 +· · ·+αn , β) with parameters α1 +· · ·+αn and β. Thus we recapture in the previous corollary a well known result in Probabilities. References [1] H. Jasiulewicz and W. Kordecki, Convolutions of Erlang and of Pascal distributions with applications to reliability, Demonstratio Math., 36:1(2003), 231-238. [2] N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions -1, Wiley, New York, 1994. [3] W. Kordecki, Reliability bounds for multistage structure with independent components, Stat. Probab. Lett., 34(1997), 43-51.

ON THE CONVOLUTION OF GAMMA DISTRIBUTIONS

211

[4] M. V. Lomonosov, Bernoulli scheme with closure, Problems Inform. Transmisssion, 10 (1974), 73-81. [5] A. M. Mathai, Storage capacity of a dam with gamma type inputs, Ann. Inst. Statist. Math., 34(1982), 591-597. [6] A. Sen and N. Balakrishnan, Convolutions of geometrics and a reliability problem, Stat. Probab. Lett., 43(1999), 421-426. D´epartement de Math´ematiques, Universit´e Cadi Ayyad, Facult´e des Sciences-Semlalia, Bd. du Prince My. Abdellah BP 2390, Marrakech, Maroc (Morocco). E-mail: [email protected]