0) or negative binomial (s, 8 < 0) into powers of u = e/( 1 + (3)’ (r positive integer, t9> 0 or r, 8<0) and setting p= 0(1 +a)-‘. Restrictions on the parameter space of these distributions were described by Nelson (1975). Consul (1974) considering a two urn model with predetermined strategy derived the probability function
h(x; s, r, n) =
n
0 X
s(s+rx-
l),_t(m+r(n-x)),_,
,
(S+m+m),
x=0,1,2
,..., n, (1.3)
which was called generalized or quasi hypergeometric distribution I when s, r, m are positive integers and generalized or quasi Polya distribution I when s,r,m are negative real numbers (see also Janardan (1975)). Another generalization of the hypergeometric and Polya distributions to this direction was obtained by Consul and Mittal (1975) by considering a four urn model with predetermined strategy. The probability function of these distributions may be written in the form P(X; s, r, n) =
n 0X
.s(s+rx-l),_lm(m+r(n-x)),_,_l (s+m)(s+m+m-
1),-r
,
x=0,1,2
)..., n. (1.4)
This distribution was called generalized or quasi hypergeometric distribution II when s, r, m are positive integers and generalized or quasi Polya distribution II when s,r, m are negative real numbers. Janardan and Schaeffer (1977) obtained this distribution as a model for voting behavior in two party situations. The discrete class of Gould series distributions introduced in this paper (Section 2) is an extension of the factorial series distributions introduced by Berg (1974,1975), to the direction of the preceding generalizations. Therefore it provides a unified approach to the study of these distributions. The generating function and factorial moments of this class of distributions are derived and it is pointed out that the distribution (1.2) is indeed a generalized general binomial (binomial or negative
Ch.A. Charalambides / Gould series distributions
17
binomial) distribution (GGBD) (Section 3). Finally it is shown that within the class of Gould series distributions, the GGBD is the only distribution which is closed under convolution (Section 4). 2. Gould series distribution 2.1.
Definition and notation
Consider a positive function A@; r) of a parameter s which may depend on an additional parameter r and assume that it admits a Gould series expansion: A@; r)=
E a(x; r)s(s+rxx=0
I),-,,
(s,r)E%xRo,
with So= {s: Is/
(2.1)
a(x; r), x= 0,1,2, . . . , in-
1),-r
=s(s+rx-l)(s+rx-2)*..(s+rx-x+1),
x=1,2,...,
Go@; r) = 1,
discussed by Gould (1962) in connection with a generalization of the binomial formula, were named after Gould by Roman and Rota (1978). The coefficients in the expansion (2.1), on using the (symbolic) formula ES=
E s(s+rx-
1),-r -$ klE-‘Y
(2.2)
x=0
where E is the displacement (shift) operator, d = E- I the difference operator and the Abel-difference operator (Roman and Rota (1978)), may be obtained as
AE-’
a(x; r)= -$ (dEp’)XA(u;
r)luzo,
rERo.
(2.3)
Restricting the parameter space So x R, to Sx R for which the terms of the expansion (2.1) are nonnegative (for example we may have a(x; r) r 0 and 0 I s< so, Ozsr
x=O,1,2,...,(s,r)~SxR, (2.4)
satisfies the properties p(x;s,r)zO,
of a probability
function.
x=0,1,2
,...,
~~op(x;$l)=l,($r)~S~R,
18
Ch.A.
Charalambides
/ Gould
series distributions
Definition 2.1. A family of discrete distributions ( p(x; s, r), (s, r) ES x R} is said to be a Gould series distribution (GSD) family with parameters s, r and series function A@; r) if it has the representation (2.4) with the series function satisfying the condition (2.1). As a convenient shorthand notation for the expression “the random variable has a GSD with series function A(s; r)” we shall write X- GSD(A(s; r)).
X
Remark 2.1. The range of x in (2.4) as in the cases of power series distributions (PSD’s) and factorial series distributions (FSD’s) may be reduced. Thus we may have T={xO,xO+l,..., x,+x,l}, x,,rO, x,2 1, a(x; r)>O, XE T. Moreover note that the truncated versions of GSD’s are also GSD’s in their own right. 2.2.
Fluctuations of sums of random variables
or in particular independent and idenLet Y,, Y2, . . . , Y,, . . . be interchangeable tically distributed random variables taking on the nonnegative integral values andN,=Y,+Yz+...+Y,, n=l,2,.... Further forsll, letD, {O,r,2r )... }, rzl, be the smallest n such that N, = n -s; if there is no such n then 0, = m. Note the random variable 0, can be interpreted as (i) the duration of the game in the theory of games of chance, (ii) the busy period in queueing processes, (iii) the time of emptiness in dam and storage processes. Then, the probability function of X= (D,-.s)/T, on using the relation (Takacs (1967), Chapter 2) 4D,=n)=;P(N,,=n-s) may be obtained
as
P(X=x)=
-s+rxs P(N,+,=rx).
(2.5)
By considering Z,, = Y,/r, n = 1,2, . . . , to be (i) a Bernoulli sequence and (ii) a Pascal sequence, the probability function of X is obtained in the form of (1.2). (i) Bernoulli sequence. Assume that Z,,, n = 1,2, . . . , are independent and identically distributed zero-one Bernoulli random variables with probability of success p. In this case the sum 2, + 2, + ... + Z, = N,/r has a binomial distribution with probability function
ek P(r~‘Nn=k)=(l+8)-“~(n)k, Hence
by (2.3,
the probability
function
P(~=x) = (I+ 8)-s S
=s+rx
k=0,1,2,...,n,e=p/(l-p). of X is given by
eyi -t 8)-rx x! s(s+ rx -
s+rx
( > x
pyl-p)S+rX--X,
l),_ , x=0,1,2
,...,
p=e/(l+e),
19
Ch.A. Charalambides / Gould series distributions
where
s, r are positive
integers
and O
1,1/r},
which
is the generalized
binomial distribution (1.2). (ii) Pascal sequence. Suppose that Z,, n = 1,2, . . . , are independent and identically distributed random variables with P(Z, = z) =pq’, z = 0, 1,2, . . . , p + q = 1. Then the sum Z, + Z, + ... +Z, = N,/r has a negative binomial distribution with probability function
=(l
Hence
by (2.9,
-9)”
the probability
g
function
(-q)X;,-q)rX
P(X=x)=(l-q)
(-n)k,
k=0,1,2
)....
of X is given by
(-s)(_s_rx_
I),_,
=~x~-“;“)(~)‘(~)._‘~,.
where -s, -r< 0 and l/(-r) binomial distribution (1.2).
< -q/( 1 -4)
is the generalized
negative
Remark 2.2. Poisson arrivals and exponential service times. Consider
a queueing process beginning with s customers and assume the customers arrive in batches of size r in accordance with a Poisson process of intensity A and served by a single server. Further suppose that the service times are independent and identically distributed negative exponential random variables with mean l/p and independent of the arrival times. The stochastic law of the busy period can be obtained by using the preceding model of a Pascal sequence. Let Z,,, n = 1,2, . . . , be the number of batches of customers arriving after the (n - I)-st departure and fore the n-th departure. Then P(Z, = n) =pqz, z = 0,1,2, . . . , with p =&(A +p), q = A/@. +,D) and using the result of preceding model the probability function of the number X of batches arrived before the queue first vanishes is obtained as
P(X=x)
=
which for s = r deduces
s+r~+x(S+~+X)(&)“(&-r;)lirx to the distribution
(1.1) obtained
by Takacs.
Several applications of particular members of the GSD have already been discussed. Here two models where the pertinent distribution turns out to be a GSD are described.
Ch.A. Charalambides / Gould series distributions
20
(a) Busy period in queueing processes. Consider a queueing process with s customers and assume the customers arrive in batches of r and are a single server. Let Z, be the number of batches of customers joining during the n-th service, n= 1,2, . . . , and suppose that the random z, ) z2, . . . ) z,, . . . are either interchangeable or in particular independent
beginning served by the queue variables and iden-
taking on the nonnegative integral values (0,1,2, . . . }. Moreover B,=Z,+Z,+ ... + Z,, the total number of batches of customers joining the queue up to the n-th service, n = 1,2, . . . , C, the number of customers served and X= (CO - s)/r the number of batches of customers (who at their arrival find the server busy) served before the queue first vanishes. (b) Time of first emptiness in dam and storage processes. The preceding model can be restated in the language of discrete dam and storage processes. In this case Z, represent the water quantity which flows into the dam at time n = 1,2, . . . or the numbers of batches of commodity coming into the warehouse at epoch n = 1,2, . . . and C, the time of first emptiness. tically distributed
3. Generating 3.1.
functions
and moments
Probability generating function
The probability generating function From (2.3) and (2.4) we have
(pgf) of the GSD may be obtained
as follows.
G(t) = c p(x; s, r)t” x=0 =[A@;
r)]-lXtos(s+rx-
which on using the generating
function
1),-r
i
tX[(dE-‘)X[A(u;
of the Gould polynomials
r)l,,,] (Roman
and Rota,
1978)
Xgos(s+rxwhere w = h-‘(u)
1),-r
is the inverse
5
=[l +h-I(o
(3.1)
of u = h(w) = w( 1 + w))‘, may be written
G(t) = [A@; r)]-‘[1 +h-‘(tdE-‘)]SA(u; 3.2.
Factorial moment generating function
The factorial moment generating function formally as follows. From (3.2), we have
F(t)=G(t+ where
w = h-‘((I
I)= [A@; r)]-‘[l
+ l)dE-‘)
implies
(fmgf)
of the GSD may be obtained
l)dE-‘)ISA(u; = w(1 + w))’ and
as
(3.2)
and moments
+h-‘((t+
(t + l)dEr
r) luzo.
formally
r)luEo
21
Ch.A. Charalambides / Gould series distributions
t=(w/Ll)[l Putting z=(w/A)h-‘((t
+(A/E){(w/d)-
1, Q=A/E, + l)AE-‘)/A
1}1-‘- 1.
we get t=(l
+z)(l +Qz)-‘-
1 and
= 1 +f-‘(t)
where z = f -l(t) is the inverse of t =f(z) = (1 + z)( 1 + Qz))‘F(t) = [A@; r)]-‘[1
+ QJ-‘(t)lSESA(u;
1. Therefore
r) luEo.
(3.3)
In order to express the factorial moments of the GSD in a close form consider the Bell partition polynomials (Riordan (1968), Chapter 5) Y, = Yn(fg,,fg2, . . . ,fg,) defined by
yi=c “!?.k,! k’k! I-
($($)“i...($
n=1,2 )...,
Y,=l.
2.
(3.4) where the summation is extended over all partitions of n, that is over all k,?Osuchthatk,+2k,+...+nk,=n; k=k,+kz+...+k,isthenumber k,,k,,..., of parts of the partition. The special case fk = (&, k = 1,2, . . . , s a real number, of the Bell partition polynomials denoted by P(‘)(gr , g2, . . . , g,), are called potential polynomials (Comtet (1974), Chapter III), hive generating function
g,)
2 =[l + {g(t) -mlls>
and satisfy the recurrence relation (Charalambides,
1977)
tk
dt) = kgogk z
3
(3.5)
i PCS) @I,...,gn+1)= ,-,(:)~-~)gk,,ql.',(g,,...,g,,), n+l
n=0,1,2 )..., pt’=1.
(3.6)
and P”‘(cgl ,...) c”g n)=c”P’qg, n n
)..., g ” ) .
(3.7)
From (3.3), (3.5) and
it follows that (3.8)
g = Q d’?-‘(t) m dt”
t=o
,
m=l,2
,..., n.
The derivatives of the inverse series f-‘(t) at zero can be expressed in terms of the derivatives of the series f(t) at zero,
Ch.A.
22
f
k
=
Charalambides
dkfW dtk
/ Gould series distributions
=(-i)k-1(r+k-2)k_rQk-1[k-(r+k-1)Q], t=o k= 1,2 ,..., m,
as follows.
From
the Lagrange
d”‘f -l(t) dtm
=I=0
inversion
d”-’ dt”-’
f(t) (>I
t
formula
we have
-” t=o
and using (3.5), we get
d"'f ~’ (t)
,
dt"
m=l,2
)..., n,
and gm=Q(l-rQ)-“*P~~/(~t,c~
m=1,2
,..., c,~,),
,..., n,
(3.9)
with Ck=(-l)k(r+k-
From
-rQ)-‘[(k+
l)kQk(l
(3.8), (3.9) and (3.10),
l)-(r+k)Q]/(k+
on using (3.7) it follows
1).
(3.10)
that
~u(,,(~,~)=[A(~;r)l~‘(-1)“Q”(1-rQ)-”P~)(al,a2,...,a,)ESA(u;r)I.=o (3.11) where a,=-Pi-f/(b,,b?,
,...,
(3.12)
b,,_,)
with bk=(r+k-
I),+(1 -rQ)-‘[(k+
l)-(r+k)Q]/(k+
1).
(3.13)
The computation of the factorial moments up to order n = 10 is facilitated by the table of the Bell partition polynomials (3.4) with fk = (Qkin Riordan (1968) and up to order n = 12 by the table of the partial Bell polynomials Bn,k(gr,gl, . . . , g,) in Comtet (1974) and the relation
P’%l~ g, . n
(3.14)
. . ..g.)=k~l(s)k~.,k(gl.g2.....g~).
For the computation of the factorial moments of higher order the recurrence lation (3.6) can be used. The first two factorial moments of the GSD may be obtained as il((s, r) = [A@; r)]-‘xl(E-
f-A)-‘FA(u;
.I+~)(s,r) = [A@; r)]-’ {s(s+ r)A2(E+s(r-
r) luzo
re-
(3.15)
rA)-2
l)A2E(E-rA)~3}ESA(u;
r)luzo.
(3.16)
23
Ch.A. Charalambides / Gould series distributions
3.3. Examples (i) Generalized binomial and negative binomial distributions.
The series func-
tion of these distributions is given by A@; r) = (1 + Q)” where s is a positive integer and 6’> 0 (generalized binomial) or s is a negative real number and B < 0 (generalized negative
binomial).
Since
(AE-‘)X(I +e)“/.=,=[e(l it follows
+by]“,
x=0,1,2
)...,
from (3.2) that I
= (I +
I
=
e)-ql
+ ekqt)]”
where
h?(te(i
+ e)y)/e
is a probability generating function and therefore (1.2) are generalized binomial or negative binomial distributions in the sense of Johnson and Kotz (1969). The generalizing distribution may be derived as follows. From (3.1) and since Lls(s+rx-
1),-r Is=e=(s+r),(s+rx-
A[1 +h-‘(v)]“ls=o=h~l(u)[l
l)x_~ls=~=(rx)x_,,
+A-‘(u)]S/,=,=h-‘(o)
we get
K’(u)=
: (TX),_1 ;
x.
X=1
,
e”-‘(i
+e)yrX,
py1
-py-X+l,
H(i)=
1,
and
where
r is a positive
integer
and O
x= 1,2 )...,
or r is a negative
(3.17) real number
and
l/r
f,=P(N,#i,i=1,2,...,n-l;N,=n)
where N,, = Y, + Y, + .e. + Y,, n = 1,2, . . . , and Y,, Y,, . . . is a sequence of independent and identically distributed binomial random variables with parameter (r,p), are given by
24
Ch.A. Charalambides / Gould series distributions
f,=
rp b(x; r,p)= 1-P
f
(
nyl>p”(1
-p),,_..
Returning to the generalized binomial and negative compute the mean and variance of these distributions find
binomial distributions let us using (3.15) and (3.16). We
k=O
p(s,r)=sp(l
-rp)-‘,
p=O/(l
+O),
,u~2~(s,r)=(1+t9~“{s(s+r)A2(E-rA)-2+s(rrk(l +
rk(l
l)A2E(E-rA)-3}(1
e)p2k+2E-k-2(i
+ey
+e)-2k+2E-k-2(i
+ ey
p=e/(i+e).
,u&s,r)=s(s+r)p2(1-rp)-2+s(r-l)p2(1-rp)-3,
02(sr) =P&, r) +ru@, f-1- MS,r)12=sp(l
+O)”
-p)U
- W3.
(ii) Quasi-Polya distribution I. The probability function of this distribution is given by (1.3) and therefore it is a member of the GSD’s with A@; r) = (s + m + rn), where s,r,m are positive integers (positive quasi hypergeometric I) or s,r,m are negative real numbers (quasi-Polya I). Since dk+lE-k-‘(S+m+r~),=(n)k+,(s+m+r~-k-l),_k_, the mean of this distribution,
P(S,r) =
on using
s E
(S+m+rn), s
= (s+m+rn),
Moreover
from (3.16) it follows
(3.15), may be obtained
rkAk+lE~k-‘(s+m+m),
k=O
n-l c k=O
that
rk(n)k+I(S+m+r~-k-l),_k_,
as
Ch.A. Charalambides / Gould series distributions
,q2)(.s
r)
25
=
s+m+rn-k-2 n-k-2 and 0~6, r)
=,q2)(4
4
+A%
r)
-
MS
f-II2
+
k+2 2
-
(s+::rn),,]2[
s+m+rn-k-2
( >I(
+(r-1)
:$k(s+:‘:-:-
n-k-2
>
l)12y
or
o’(s, r) =
sn s+m+rn + (s+r:rn)
:,Irkk+(s+r)(‘i +(r-
‘) s-tm+rn-k-2
1)
n-k-2
2(s+m+rn)-k-
(iii) Quasi-Polya distribution II. The probability is given by (1.4) and therefore it is a member
(.s+m)(.s+m+rn-l),_, hypergeometric
the mean
where s,r, m II) or s, r, m are negative
of this distribution,
function of this distribution of the GSD’s with A@; r) =
are positive real numbers
on using (3.19,
1
integers (positive quasi (quasi-Polya II). Since
may be obtained
as
Ch.A.
26
~6, r) =
Charalambides
/ Gould series distributions
sn!
s+m+rn-k-l
(s+m)(s+m+rn-1)
n-k-l sfmfrn-k-2
ns Ms, r) = SSrn’ The second
factorial
moment,
on using
(3.16), may be obtained
as
sn! ~(2#
r) =
(s+m)(s+m+rn-l),_,
P&,
r) =
n(n-
l)s(s+2r-
n!s
1)
(s+m)(s+m+rn-1)
+ (.s+m)(s+m+rn-l),_,
n-3
. k;, rk[(s+r)+(r-
l)(k+
l)]
“‘“n+~~~e2).
Therefore ns a’@, r) = s+m
n(n-
-
+
l)s(s+2r-
n!s
1)
(s+m)(s+m+rn-1)
+ (s+m)(s+m+rn-l),_,
n-2
. kg,rk[(s+r)+(r-
4. A characterization
l)(k+
of the generalized
l)]
““,‘~~~-‘).
general binomial
distribution
Note first that for a GSD with a(0; r) =A(O; r) 20 we may always have a(0; r) = A(0; r) = 1, since otherwise by putting a*@; r) = a(x; r)/a(O; r) and A*(s; r) = A(s; r)/A(O; r) we obtain a*(O; r) =A *(O; r) = 1. Let XI and X2 be independent random variables with Theorem 4.1. Xi - GSD(A(s;; r)), si nonnegative integer and r positive integer or si nonpositive real number and r negative real number, i= 1,2, A(0; r) f0, AA(u -r; r)j,,=,+O, and Y= X, +X2. Then Y- GSD(B(s; r)) if and only if Xi - GGBD(asi, ar), i = 1,2 with a an arbitrary constant. Proof. Using the total probability probability function of the sum
theorem and the independence of X, and X2, the Y = XI +X2 may be obtained as
21
Ch.A. Charalambides / Gould series distributions
y=o,1,2
(4.1)
)....
Hence P( Y= 0) = [A@,; r)A(s,; r)]-,,
P(Y= l)= [A@,; r)A(s,; r)]-‘a(l;
r)(s, +sd
and since
P(Y=y)=[B(s;r)]-‘b(y;r)s(s+ry-l),_,, it follows
(4.2)
y=O,l,L...,
that
b(1; r)s=a(l;
B(s; r) =A(s,; r)A(s2; r), Therefore,
on
using
the
r)(s, +s,).
~(1; r)=dA(u-r;
assumption
(4.3)
r)lu=OfO,
whence
6(1; r)#O,
c(r)=a(l;
s = c(r)(s, f sd,
r)/b(l; r)>O
(4.4)
and
B(c(r)(s,+sz);r)=A(s,;r)A(s,;r),
sj=0,1,2 or
The solutions given by
of this functional
equation
A(si; r) = (1 + @‘I, where 8= e(r)>0 numbers. Thus
,..., r=l,2
,...
~~50, r
satisfying
A(0; r) = 1 are
the condition
B(s; r) = (1 + f9)s’c(T,
if s, r are positive
integers
and f3= e(r)50
if s, r are negative
real
a(l;r)=dA(u-r;r)l.=o=e(l+e)-‘, 6(1;r)=dB(u-r;r)/.=,=(l+e)-‘~c~‘~[(l+e),~~~’~-l], and using the second
(s, +s,)(s, +s,+ryit follows
ioL
that c(r) = 1 and s=s,
of (4.4) it follows l),_,
= i
’
X=00 x
+s2. Since
s,(s,+rx-l),_,s2(s2+ry-rx-1)y_X_,
from (4.1), (4.2) and the first of (4.3) that a(x; My-x;r)-
which by virtue
(z)
b(y.r) 7 ] s,(s,+rx-l),_,s2(s2+ry-rx-l),_,_,=O
of the orthogonality
a(x; r)a(y-x;
r)=
of the Gould
y b(y;r), 0X
x=0,1,2
polynomials
is equivalent
,..., y, y=O,1,2
,....
to
28
Ch.A.
Putting
Charalambides
/ Gould series distributions
x = 0 and since a(0; r) = 1 we get b(y;r)=a(y;r),
y=o,1,2
)...)
and a(x; r)a(y-x;
r)=
Y
0
4Y;
X
r), x=0,1,2
which since a( 1; r) = 13(1 + 8)-’ is equivalent a(x;r)=aX[8(1+8)-‘jX/x!, and hence Xi - GGBD(as,;
,...) y, y=o,
1,2 )...)
to
x=0,1,2
)...)
at-), i = 1,2. Moreover
or), s = s1+ s2.
Y- GGBD(m,
References Berg, S. (1974). Factorial Statist.
series distributions
with applications
to capture-recapture
problems.
Stand.
J.
1, 145-152.
Berg, S. (1975). A note on the connection power
series distribution.
Charalambides,
Stand.
between
Actuar.
factorial
series distribution
and the zero truncated
J. 233-237.
C.A. (1977). On the generalized
discrete distributions
and the Bell polynomials.
Sankhya
Ser. B. 39, 36-44. Comtet,
L. (1974). Advanced
Consul,
P.C. (1974). A simple urn model dependent
Combinatorics,
Reidel,
Dordrecht. upon predetermined
strategy.
Sankhya
Ser. B 36,
P.C. and S.P. Mittal (1975). A new urn model with predetermined
strategy.
Biometrische
391-399. Consul,
Z. 17,
67-75. Consul, P.C. and L.R. Shenton (1972). Use of Lagrange expansion probability distributions. SIAM J. Appl. Math. 23, 239-248. Dwass,
M. (1967). Poisson
Gould,
H.W.
(1962).
Mathematical Haight,
F.A.
Notes,
recurrence
Congruences
times. J. Appl.
involving
Probab.
for generating
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