Download on the convolution of gamma distributions

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.
208
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)
210
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.
Département de Mathématiques, Université Cadi Ayyad, Faculté des Sciences-Semlalia, Bd. du
Prince My. Abdellah BP 2390, Marrakech, Maroc (Morocco).
E-mail: [email protected]