Gould series distributions with applications to fluctuations of sums of random variables

Gould series distributions with applications to fluctuations of sums of random variables

Journal of Statistical Planning and Inference 15 14 (1986) 15-28 North-Holland GOULD SERIES DISTRIBUTIONS WITH APPLICATIONS TO FLUCTUATIONS OF ...

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Journal

of Statistical

Planning

and Inference

15

14 (1986) 15-28

North-Holland

GOULD SERIES DISTRIBUTIONS WITH APPLICATIONS TO FLUCTUATIONS OF SUMS OF RANDOM VARIABLES Ch.A.

CHARALAMBIDES*

Statistical Unit, University of Athens, Panepistemiopolis, 15700 Athens, Greece Received

10 September

1984; revised

Abstract: A two-parameter by expanding Gould

a suitable

series distribution

particularly cesses.

distributions

function.

under

series distributions, polynomials

and

the factorial

It is pointed distribution

variables

moments

of Consul

of the Gould

and Jain is justified general

A and

of games of chance,

in dam and storage

out that the name of the generalized

it is shown that the generalized

is the only member

generated

is discussed.

random

of the game in the theory

and (iii) the time of emptiness

binomial)

Finally

under certain mild conditions,

Gould

1985

of sums of interchangeable

function

in close forms.

or negative

15 September

into a series of Gould

the duration

processes

generating

are obtained

(binomial

of its generating

of(i)

in queueing

received

distributions,

function

occurs in fluctuations

The probability

binomial

class of discrete parametric

as the distribution

(ii) the busy period

manuscript

binomial

of the Gould series distributions

proseries

general

by the form distribution,

which is closed

convolution.

AMS Subject Classification: Primary

63ElO;

Key words andphrases: Gould polynomials; distributions; negative

Quasi

binomial

Polya

distributions;

Secondary Generalized

05AlO. (Quasi) binomial

Characterization

and negative

of the generalized

binomial

binomial

and

distributions.

1. Introduction Takacs (1962), as an application of a generalization of the ballot problem, considered a queueing process where customers arrived in batches of size r in accordance with a Poisson process of intensity A and served individually by a single server. The service times were assumed independent and identically distributed negative exponential random variables with mean l/p and independent of the arrival times. Then the probability function of the number X of batches of arrivals after the queueing process starts (i.e. not including the one that starts the busy period), before the queue vanishes was derived as b(x; r, B) = .+,:+*(l+~+x)8,(1-o)““, As a special * Presently

case of a generalization

at Temple

0378.3758/86/$3.50

University,

6

Philadelphia,

1986, Elsevier

Science

8=&,x=0,1,2 of a two coin tossing PA 19122, USA.

Publishers

B.V. (North-Holland)

,.... problem,

(1.1)

Mohanty

16

Ch.A.

Charalambides

/ Gould series distributions

(1966) obtained an analogous distribution for the case of a queueing process beginning with s customers. The distributions obtained by Takacs and Mohanty are special cases of the generalized binomial or negative binomial distributions with probability function b(x; s, r, p) =

A--

s+rx

s+rx

( > x

pX(l -p)s+rX-x,

x=0,

1,2, . ..)

(1.2)

with parameters (i) s, r positive integers and 0 < p < min{ 1,l /r}(generalized binomial) or (ii) s, r< 0 and l/r 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/ and coefficients dependent of the parameter s. The polynomials G,(s; r)=s(s+rx-

(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

discrete

generalized

4, 605-608.

sums of binomial

coefficients

and a formula

of Jensen.

May 1962, 400-402.

(1961). A distribution

Jain, G.C. and P.C. Consul

analogous

(1971). A generalized

21, 501-513. Janardan, K.G. (1975). Markov-Polya

to the Bore]-Tanner. negative

Biometrika

48, 167-173.

binomial

distribution.

SIAM

urn model with predetermined

strategies.

Gujarat

J. Appl. Math. Statist.

Review

2, 17-32. Janardan,

K.G.

and D.J.

Schaeffer

(1977). A generalization

of Markov-Polya

sions and applications. Biometrical J. 19, 87-106. Johnson, N.L. and S. Kotz (1969). Distributions in Statistics:

distribution,

Discrete Distributions.

Miffin,

its extenNew York.

Mohanty, S.G. (1966). On a generalized two-coin tossing problem. Biometrische Z. 8, 266-272. Nelson, D.L. (1975). Some remarks on generalizations of the negative binomial and Poisson distributions.

Technometrics

17, 135-136.

Riordan, Roman,

.I. (1968). Combinatorial Identities. Wiley, New York. S.M. and G.C. Rota (1978). The umbra1 calculus. Adv.

Takacs,

L. (1962). A generalization

of the ballot

problem

J. Amer. Statist. Assoc. 57, 327-337. Takacs, L. (1963). The stochastic law of the busy period J. Math. Anal. Appt. 6, 33-42. TakBcs, L. (1967). Combinatorial methods

in the Theory

Math.

27, 95-188.

and its applications

in the theory

of queues.

for a single server queue with Poisson of Stochastic

Processes.

Wiley,

input.

New York.