Bivariate distributions generated from Pólya-Eggenberger urn models

JOURNAL

OF MULTIVARIATE

35, 48-65 ( 1990)

ANALYSIS

Bivariate Distributions from P6lya-Eggenberger

Generated Urn Models

ALBERT W. MARSHALL* University Western

of British Columbia and Washington CJniversitj

AND INGRAM OLKIN+ Stanford Communicated

University by the Editors

In several classic papers, P6lya and Eggenberger used urn models to generate distributions that could be used to model contagion processes. Three bivariate versions are investigated together with their limiting distributions. In this way a large class of bivariate distributions (beta, Pareto, gamma, negative binomial, Poisson, binomial) is obtained. These derivations also show connections between the various distributions. In particular, some limiting distributions are shown to be upper Frtchet bounds, and some distributions are associated. ‘e. 1990 Academic Press, Inc.

A. INTRODUCTION From an urn containing a red and b black balls, a ball is removed; its color is noted and the ball is returned to the urn along with s additional balls of the same color. The experiment is then repeated using the newly constituted urn, and in this way a sequence of trials is performed. This briefly describes the well-known Pdlya-Eggenberger urn scheme [6, 141. For the Polya-Eggenberger urn scheme, the number of red balls drawn in the first n trials is said to have a Pdlya-Eggenberger distribution. The urn Received October 27, 1989; revised February 14, 1990. AMS 1980 subject classifications: 6OE05, 62E99. Key words and phrases: bivariate binomial distribution, bivariate negative binomial distribution, bivariate Poisson distribution, Dirichlet distribution, bivariate gamma distribution, bivariate beta distribution, bivariate Pareto distribution. Frechet bounds, associated random variables. * Research supported in part by the National Science Foundation and by the Natural Sciences and Engineering Research Council of Canada. + Research supported in part by the National Science Foundation.

48 0047-259X/90 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.

BIVARIATE DISTRIBUTIONS FROM URN MODELS

49

model and the Polya-Eggenberger distribution have been used extensively in studies of contagion (see, e.g., [l, 161). In addition to the number of red balls drawn in the first n trials, the number of black balls drawn before the rth red ball is of interest; this number is said to have a negative Pblya-Eggenberger distribution. We have already considered [17] bivariate versions of sampling with replacement (s = 0), in which case successive trials are independent. One purpose of this paper is to do the same for sampling according to bivariate Polya-Eggenberger urn models or, more generally, for successive trials that are exchangeable. However, in this paper, only Pblya-Eggenberger distributions are considered; those that stem from negative PolyaEggenberger distributions will be treated elsewhere. Distributions which are weak limits of univariate Polya-Eggenberger distributions have been identified by Eggenberger and Polya [7]. In Sections 1 and 2 of this paper we derive the distributions and give some insight into why particular limiting distributions arise. Bivariate versions are obtained in Sections 3 and 4 using three different bivariate urn models. Only notational problems stand in the way of extensions to higher dimensions; while these problems are not serious, we fear that they could obscure the simplicity of the underlying ideas. Consequently we have elected to treat only the bivariate case in this paper. Some properties of bivariate Polya-Eggenberger distributions are given briefly in Section 5.

1. UNIVARIATE

P~LYA-EGGENBERGER

DISTRIBUTIONS

For sampling according to the urn scheme described above, let Zi= 1 if a red ball is drawn on the ith trial, and Zj= 0 otherwise. A remarkable feature of the Polya-Eggenberger urn scheme is that 2,) Z2, ... are exchangeable. When s >O there is no limit to the number of trials that can be performed. Then, according to de Finetti’s theorem [9, p. 2281, this means that the joint distribution of Z,, ,.., Z, has a mixture representation P(Z, = i,, .... Z,=i,)=J*’

fi

0 j-1

t?~(l--tJ)‘-~dG(8),

i,=Oor For the Polya-Eggenberger distribution with density

1,

j = 1, .... n.

urn scheme, the mixing distribution

(1.1)

G is a beta

(1.2)

50

MARSHALL

AND

OLKIN

To simplify notation, we assume that s = 1 and compensate by allowing a and b to take non-integer values. If X= Cr= i Zi denotes the number of successes (red balls) in the first n trials, then it follows from (1.1) that

P(x=x)‘j;(:)

W(l -@+*dG(8),

x = 0, 1) .... n.

In the special case of the Polya-Eggenberger

urn scheme (s > 0),

(1.3)

h(x 1a, 6, n) := P(X= x)

(1.4)

so = l n

=

0 x

eyi -e)-

e-1(1

-epp+,

0

b)

x=0,

n B(x + a, n -x + b)/B(a, b), X

de

l,..., n. (1.5)

Here, X is said to have a P6lya-Eggenberger distribution (see, e.g., [2; 8, p. 119; or 13, Chap. 9, Sect. 4). A distinction of the case s < 0 is that removal of t = --s balls from the urn can occur at most n = [(a + b)/t] times, so that only finitely many trials can be performed. In fact our description of the Pblya-Eggenberger urn model is not complete if it is possible for the urn to contain a positive number less than t of balls of some specified color. To avoid this possibility we make the following Assumption. b>(n1)t.

Either

t= 1

(and

a+ ban),

or

a> (n-

1)t

and

Under this assumption, Z, , .... Z, are exchangeable, but de Finetti’s theorem does not apply and the distribution of X= x7= 1 Zi has no mixture representation such as (1.4). However, elementary calculations show that

P(X=x)=

(--

1

n (x)r(~-~+~~~(~‘:+x+l) r(;+1)r(4+1) .

r(a:h

f lJ

,

x=O,L...,n.

Again it is possible to simplify notation by taking t = 1 and allowing a and

BIVARIATE

DISTRIBUTIONS

FROM

URN

51

MODELS

b to take non-integer values. Then, the above assumption requires that either a and b are integers or a, b > n - 1; under these conditions,

hp(xIa,

b, n) :=P(X=x) n

1) l)T(b-n+x+l)’

r(a+b-n+

T(a+ l)T(b+

= 0 x T(u-x+ x = 0, 1, .... n.

T(u+b+l)

1) ’ (1.6)

Notice that when a and b are integers, (1.6) is just the familiar hypergeometric distribution, because s= - 1 is the case of sampling without replacement. 2. THE UNIVARIATE

LIMITS

Eggenberger and Polya [7] state that the only limiting distributions of (1.4) are the binomial, negative binomial, beta, Poisson, gamma, normal, and degenerate distributions. They show how each of these limits can be obtained, but the way they discovered their results is not apparent. The purpose of this section is to briefly review the results of Eggenberger and Polya and to show how they can be obtained by pairing the limits of the binomial distribution with the beta distribution in (1.3); insight from the univariate case is applied later in two dimensions. 2.1. Binomial Limit, s = 1 From a representation of the form

p(X=x)=~((:) P( 1 -

,)‘-,

dG,(8)

it is apparent that if G, converges weakly to the distribution

degenerate at

p, then

lim P(X=x)=

m-m

n p”(l-p)“-“. 0X

In case G( .I a, b) is a beta distribution given by (1.2), EO = u/(u + b), Var 0 = ub/((u+ b)’ (a + b + 1)). Thus, if a, b + cc in such a way that u/(u + b) = p, then G( . I a, 6) converges weakly to the distribution degenerate at p. Consequently, if P(X= x) is given by (1.4) or (1.5), lim o,b+m a/(o+b)=p

52

MARSHALL

AND

OLKIN

2.2. Beta Limit, s = 1 The representation

(1.3) has the form

P(X=.)=

j,‘P(~Z;=.xIB)dG(B), 1 where Zi are independent and identically distributed with EZi= 8. Consequently by the law of large numbers, lim P(f
jO1P(~~Zi<,lR)dG(R)

or,

=

I I0

I i.x>s; We)

For the case of the Polya-Eggenberger tion.

= G(x).

distribution,

G is a beta distribu-

2.3. Negative Binomial Limit, s = 1 It is well known that if X has a Poisson distribution with random parameter A having a gamma distribution, then unconditionally, X has a negative binomial distribution. Thus a negative binomial limit can be expected from (1.3) if the binomial integrand converges to a Poisson limit and if G converges to a gamma distribution. Suppose that 0 has the beta density of the form (1.2) with nb in place of b, and let A = n@. Then it is easily verified that ;i~t P(~Gi.)=j~buwu~le~bi’dw, A 2 0, r(a) so that A has a limiting gamma distribution with shape parameter a and scale parameter b. In (1.4), replace b by bn, make the change of variable A = n0, and let n + 0~)to obtain

(2.1) x=0,1

which is a negative binomial distribution.

)...)

(2.2)

BIVARIATE DISTRIBUTIONS FROM URN MODELS

53

2.4. The Remaining Limits, s = 1

It is now clear that the normal, Poisson, and gamma distributions are limits of the Polya-Eggenberger distribution (1.4) because these distributions are limits of the limiting distributions already obtained. Note for later reference that the negative binomial distribution of (2.1) has a Poisson limit as a -+ cc while is fixed. This is easily verified directly from (2.2) or from the mixture representation (2.1). With the substitution a/b = 1*, the gamma distribution of (2.1) has mean p and variance p/b which converges to 0. Thus (2.2) has the limit e-q/P/x!), x=0, 1, ... . 2.5. The Case s < 0 The distribution (1.6) has binomial, Poisson, and normal limits; these results are familiar because (1.6) is essentially a hypergeometric distribution. It is elementary to verify directly that when P(X= x) is given by (1.6), lim o,b-+m u/(u + b) = p

3. BIVARIATE P~LYA-EGGENBERGER

DISTRIBUTIONS

3.1. The Models Three urn models are considered in this paper, each of which leads to a bivariate Polya-Eggenberger distribution. URN MODEL I. From an urn containing a red, b black, and c white balls, a ball is chosen at random, its color is noted, and the ball is returned to the urn along with s additional balls of the same color. Always starting with the newly constituted urn, this experiment is continued n times; the number x of red balls drawn and the number y of black balls drawn are noted. This urn model was proposed in its multivariate version by Steyn [18]; see also Johnson and Kotz [ 13, p. 3081, Janardan and Patil [ 121, and Dyczka [S]. URN MODEL II. From an urn containing av balls labeled (i, j), i = 0, 1, and j = 0, 1, a ball is chosen at random, its label is noted, and the ball is

54

MARSHALLANDOLKIN

returned to the urn along with s additional balls with the same label. Always starting with the newly constituted urn, this experiment is continued n times; the number x of balls with the first label 1 and the number y of bails with second label 1 are recorded. This urn model reduces to Urn Model I when a,, = a, a,, = b, a, = c, and a,, = 0. So far as we know, it was first proposed by Kaiser and Stefansky [lS], but no real theory has appeared in the literature. URN MODEL III. Here, use is made of the standard Polya-Eggenberger urn model described in Section 1, where Zi = 1 if a red ball is drawn on the ith trial and Zi=O otherwise. The numbers X=C::, Zi and Y = c;;;;-‘;+ 1 Zj are recorded. With k = 0, this model has been introduced by Hald [lo], but see also Bosch [2].

3.2. The Distributions

In the following derivation, when s > 1 it is assumed that s = 1, and when s < 0 it is assumed that s = - 1. As in the univariate case, these assumptions are without loss of generality if the numbers of various kinds of balls in the urns are not required to be integers. When s = 1, the joint probability mass function of x and y is denoted by h; when s = - 1, it is denoted by h ~. URN MODEL I. Consider first the case that s = 1. The probability of drawing x red balls, then y black balls, and finally n -x - y white balls in sequence is easily seen to be

a a+1 a+b+c’a+b+c+l

a+x-1

X

c+l u+b+c+x+y+l

c

b+y-1

b+l ‘a+b+c+x+l”’

b

“’ a+b+c+x-l’a+b+c+x a+b+c+x+y-l’a+b+c+x+y

. . . c+(n-x-y)-1 a+b+c+n-1

’

Moreover, the probability of drawing x red, y black and n -x - y white balls in any other order is the same. This means that multiplying the above expression by (Xr.y) yields h(x, y), T(a+b+c) T(a+b+C+n) =

B(a+x,b+y,c+n-x-y) B(a, b, c)

2

x, y=o,

13 .",

(3.1)

BIVARIATE

DISTRIBUTIONS

FROM

URN

55

MODELS

is an l-variable version of the where B(u,, .... Ml) = nf= 1 r(“j)/r(Cf, 1 ui) . beta function. It is easily verified that h can also be written in the form e;e;(l-8, Nx,A=J (xn,) 9

-e,)n-X-J

e’;-18:-1(1-e, -e,y X B(a,b,cl

dOIa,

where integration extends over the region 0 Q 8,, f&, 0, + e2 6 1. Thus h is a Dirichlet mixture of multinomial distributions. Under the conditions a-n+l>O, b-n+ l>O, c-n+l>O, similar arguments lead to the conclusion that for s = - 1, T(b+ 1) f(c+ 1) T(a + 1) Z-(u-x+ l)‘T(b-y+ l)‘T(c-n+x+ y+ 1) ’ URN

T(a+b+c-n+l) T(a+b+c+l)

II.

MODEL

’

x, y= 1,2, .... n.

(3.3)

For this model, with s = 1, n

4-T Y) = 12 a,x-a,y-a,n-x-y+a (

)

xB(u~~+a,u,o+x-a,aol+y-a,u,+n-x-y+a)

B(Q,l?a103 a01 >%o) = s fk m,,

ho, e,,) de,,, ho, eoh,, alo,sol,aoo)

x de,, de,, de,, 7

(3.4)

where integration extends over the region 8,,, Oio, eol 20, 0, = 1 - 8io - 8,, - 0, i 2 0, f is the bivariate binomial probability mass function given by fb

.e4,, =

boy e,,) n a,x-a,y-a,n-x-y+a

=( I

>

p, e;,- qjgl- yj& x - Y+ 0~ 2 (3.5)

and g is the Dirichlet density given by p; - lqdh

elo, eo, I 4,) u,,, 4d =

lpi21 01- lpaoo00 1

B(~l,~ a 10, ell,

e,,,

a01 3 a00 ) eol,

e,m

’

(3.6)

56

MARSHALL AND OLKIN

Under the conditions aij - n + 1 > 0, i, j = 0, 1, when s = - 1, n qa,, + 1) h-(-cy,=c1 (M,X-M,J’-M,n-x-y++ > f(a,,-cc+l) rta,,+ 1) r(%,+ 1)

x f(a,,-x+x+

ljT(a,,--?i+cr+

f(a-n+ 1) T(a+ 1) ’

~(%I+ 1) XT(a,-n+x+y-G(+l)’ x,y=l,2

1) (3.7)

3 .... n.

URN MODEL III. Again the derivation of h and h- is straightforward; here, it is convenient to use the relationship nlpk+rq

n,-k zi=l,

1

zi=x-1,

zi=y-f

1 i=q+

i=l

1

. I

(3.8)

Of course, the exchangeability follows that for s = 1,

X

of the Zi is also helpful. With these facts, it

B(a+x+y-I,b+n,+n,-x-y-k+/) B(a,

= )(x, 5

~10) O”-;;u-;‘b-’ ,

b)

d0,

(3.9)

(3.10) Here, h is a beta-mixture of bivariate binomial distributions different from those arising with Urn Model II. The bivariate binomial distribution f of (3.10) is the joint distribution of U + W and I’+ W, where U, V, and W are independent with respective binomial (ni -k, 0), (nz - k, 0), and (k, 0) distributions.

BIVARIATE

DISTRIBUTIONS

FROM

For s = - 1 and under the conditions 1 >o,

URN

57

MODELS

a - n, - n2 + 1 > 0, b-n,-n,+

T(a+b-n,-n,+k+l)

1) XT(b-n,-n2+k+x+y-1+l) T(b+

Ua+b,) (3.1;)

x=0,

1, ...) n,, y= 1,2, .... n*.

4.

THE

LIMITING

DISTRIBUTIONS

The univariate limits of Section 2 all have bivariate analogs, and they are obtainable using the methods of Section 2 with more or less obvious modifications. 4.1. Binomial Limits, s = 1 URN

MODEL

I.

If 0,) 0, have the Dirichlet distribution

appearing in

(3.2), then EO,=

VarO,=

EQ, =

a a+b+c’ a(b + c) (a+b+c)2(a+b+c+1)’

b a+b+c’

b(a + c) Var@2=(a+b+c)2(a+b+c+l)*

Ifa,b,cjcoinsuchawaythata/(a+b+c)=p,,b/(a+b+c)=p,,then both variances converge to zero, so h of (3.2) converges to f given by

.0X? Y) = ( xny)

7

P?P,y(l

x, y = 0, 1 , .-+, n. (4.1)

-P1-P2Y-x-u,

Of course, this is just the multinomial

probability

mass function.

URN MODEL II. Similarly, if h is given by (3.4) and a, 1, a,, , a,, , a, + CC in such a way that ag/(all + a,,+a,,, +a,,) = pii, i, j=O, 1, then h converges to ft., -1pll, plo, poll given by (3.5). URN

MODEL

III.

Just as in the univariate

case, if a, b + co while

58

MARSHALL AND OLKIN

~/(a+ b) = p, then h given by (3.9) converges to the probability function f of (3.10).

mass

4.2. Dirichlet Limits, s = 1 Following Section 2.2, the joint considered here.

limiting

distribution

of (x/n, y/:ln) is

URN MODEL I. Suppose first that (X, Y) has the probability mass function h of (3.2). Let W,(Z,) be the indicator random variable of the event “Zth ball drawn is red (white),” I= 1, .... n. Then X=x; W,, Y=CI; Z,. If G is the Dirichlet distribution of (3.2), then

limp

x
>

II. Let Z,, be the indicator random variable of the event “Zth ball drawn has label (i, j),” i, j = 0, 1, I = 1, .... n. Then X= Cy (Z,,,[+ Zio,J, Y = xy (Z,,,,+ Z,,,,). If G is the Dirichlet distribution of (3.6), then from (3.4), URN

MODEL

limp

X
=

)

Z s ix~s,o+e,,,~~eo,+e,,l dG(Q=G*(x,

Y);

here, G* is the distribution of (@,,+ @,i, O,,i + Oi,), where @ii, @,,, @,, have the joint density of (3.6). Neither G* nor its density have a nice form, but it is easily seen that G* has beta marginals and, if a= cli, + ai0 + then a01 + 40~ E(@,, +@,,)=-+

a

Var(B,, + @,,) =

E(C),, + Qol) = -+ (a11 + ~loN~ol+ 40) a2(a+1) ’

a

59

BIVARIATE DISTRIBUTIONS FROM URN MODELS

Var(O,, + O,,) = cov(Ql,+Q,o, Corr(O,,

(~l,+~o,)(~,o+%) &+l)

’

Q,, +QoI)=ul’u~-ulouol, a’(0 + 1)

+ O,,, O,, + O,,) =

ullaoo-ua,oaol

,

~~+~O+a+la+o

where a 1+

=a,,+a,o,

~+l=~ll+~ol,

a,+

=%+a,,,

a+,=u,+u,,.

distribution of III. Finally, consider the limiting (X/n,, Y/n,) for Urn Model III. It is easily seen from (3.8) that, with G denoting the beta distribution of (3.9), URN

MODEL

= G(min(x, y)) = min[G(x),

G(y)].

Of course, this is just the upper Frechet bound for bivariate distributions with both marginals equal to G. 4.3. Negative Binomial Limits, s = 1

As in Section 2.3, the convergence to a negative binomial distribution comes about by convergence of the multinomial or bivariate binomial to a bivariate Poisson distribution while at the same time the Dirichlet distribution converges to a gamma distribution. URN MODEL I. If 8, and 0, have the joint Dirichlet density appearing in (3.2) but with c replaced by nc, then the distribution of (nQ,, nO,) converges weakly to a bivariate gamma distribution with independent marginals. At the same time, the multinomial distribution appearing in the integrand of (3.2) converges to a bivariate Poisson distribution that is the product of its marginals. Consequently the bivariate negative binomial limit of (3.2) is just the case of independence.

60

MARSHALL AND OLKIN

URN MODEL II. The distribution (3.4) is more interesting because the bivariate binomial distribution of (3.5) has the bivariate Poisson limit:

x, y = 0, 1, ... .

(4.2)

If (O,,, O,,, O,,) have the joint density (3.6) with naM) in place of a,, then the limiting distribution of n(O,, , O,,, O,,i) is a product of independent gamma distributions, all with scale parameter a,, and respective shape parameters a,, , Q,~, and a,,. Thus the limiting negative binomial distribution obtained from (3.4) obtained by setting A,, = tI,,/n, A,, = 8,&z, and A,, = 6,,/n is

This is the distribution of U + W, V+ W, where U, I’, and W are independent with negative binomial distribution. Of course, this structure is inherited from the bivariate Poisson distribution. URN MODEL III. The limiting bivariate negative distribution obtained from (3.9) is an interesting contrast to (4.3) because the bivariate binomial distribution (3.10) though quite different from that of (3.6) has the same Poisson limiting distribution (4.2) when n, = (A,, + Ilio)k, n2 = (A,, + A,,)k, tJ= l/k, and k + co. In (4.3), the random parameters A, of the bivariate Poisson distribution are independent; from (3.9) the contrasting case arises in which the random parameters A, are proportional with probability one, i.e., their joint distribution is that of an upper Frechet bound (with marginals having different parameters). Thus the limiting negative binomial

BIVARIATE

DISTRIBUTIONS

FROM

URN

MODELS

61

distribution obtained from (3.9) with n, = (A,, + A,,)n, n2 = (A,, + &)n, k = A,, n, 8 = t/n, b replaced by nb, and n + cc is given by

(4.4)

4.4. Binomial Limits, s = - 1 For Urn Model I, the distribution h_ given by (3.3) is easily seen to have the limiting multinomial distribution limb-(x,

y)=

n 4 y,n-x-y

>

p;p;(l

-pl

-P2)n-r-~v,

when a, 6, and c + cc with a/(a + b + c) = pl, b/(a + b + c) = pz. In Urn Model II, set a = a,, + a,, + a,, + a,, a,/a = pli, and let a + 00. If h ~ is given by (3.7), then lim h-(x, n-m

y)=c r

a,x-a,

n y-a,n-x-

y--r n--x--)+a , OLPcllPO0 y+cc > PT, P-TO-

and this is the bivariate binomial distribution of (3.5). For Urn Model III, let a/(a + b) = p, while ~1,b + w in (3.1 l), to obtain lim h _ (x, y) = 1I (;)(“:I;)(;~;) which is the bivariate binomial distribution

px+‘-l(l

~p)nl+n2-k-x-y+I,

of (3.10).

4.5. Poisson Limits The Poisson limits of the Polya-Eggenberger distributions can be obtained sequentially from the binomial or negative binomial limits. Poisson limits of the distributions with binomial marginals: Model 1. Here the distribution with binomial marginals is a multinomial distribution and its Poisson limit is the case of independence. Model ZZ. The limiting bivariate binomial distribution is given by (3.5); its Poisson limit is given by (4.2). Model ZZZ. The limiting bivariate binomial distribution is given by (3.1); its Poisson limit is also given by (4.2). Poisson limits of the negative binomial. Negative binomial limits of Pblya-Eggenberger distributions are obtained only when s > 0 (s = 1) and

62

MARSHALL AND OLKIN

they have representation as Dirichlet mixtures of Poisson distributions. The bivariate Poisson distributions appearing in these mixtures are retrieved as limits when the Dirichlet distributions converge to a degenerate distribution. 4.6. Gamma Limits The gamma limits of Polya-Eggenberger distributions are obtainable sequentially either from the beta (Dirichlet) or negative binomial limits. Gamma limits of Dirichlet distributions: Model I. The bivariate Dirichlet distributions appearing in (3.2) is obtained in Section 4.2 as a limit of the Polya-Eggenberger distribution. If (A’, Y) has that limiting Dirichlet distribution with c replaced by nc, the distribution of (KY, nY ) converges to a bivariate gamma distribution with independent marginals. Model ZZ, If (Or,, OrO, O,,) has the joint density (3.6), then the joint distribution of (0 i, + 0 rO,Or, + O,, ) is the limiting bivariate beta distribution obtained with Model II. With a, replaced by naM) the distribution of (X, Y) = n(@,, + Q1,,, O,, + O,,) converges to the bivariate gamma distribution of (U + W, V + W), where U, V, and W are independent with gamma distributions having scale parameter a, and respective shape parameters a,,, a,, , a,, . Model ZZZ. Here, the limiting bivariate beta distributions is an upper Frtchet bound, so the limiting gamma distribution is also un upper Frtchet bound. Gamma limits of negative binomial distributions: Model Z. For this model the limiting negative binomial distribution, and hence the limiting gamma distribution, is just the case of independence. Model ZZ. Since the limiting negative binomial distribution is the distribution of U + W, V+ W, where U, V, and W are independent, the same is true of the limiting gamma distribution. Model ZZZ. The problem of determining the gamma limit of (4.4) is unsolved.

5. MOMENTS

AND PROPERTIES

This reference section briefly outlines some results about the various Polya-Eggenberger distributions. The univariate case. If X has the probability

mass function (1.4) where

BIVARIATE

DISTRIBUTIONS

FROM

URN

63

MODELS

s = 1, or (1.6) where s = - 1 (in which case a + b 2 n), it is easily verified that EX=na

VarX=n

a+b’

ab(u + b + sn) (a+b)* (a+b+s)’

s= +1.

(5.1)

In case s = 1, it is easy to verify that for a, b 3 1, h(x + 11a, b, n)/ h(xl u, b, n) is decreasing in x = 0, 1, .... n; hence h( .I a, b, n) is a Polya frequency function of order 2 when a, b 2 1. The bivuriute case: Model I. If the vector (X, Y) has the probability mass function given by (3.1) or (3.2), then h( . I a, b + c, n) and h( .I b, a + c, n) given by (1.4) are, respectively, the marginal probability mass functions of X and Y. Note that E( Y 1X = x) is the expected number of white balls drawn in the (n - .u) trials resulting in a non-red ball, so E( YI X= x) = (n-x)

b/(b + c).

(5.2)

For this model, Corr(X,

Y) = - [ub/(u + c)(b + c)]“‘.

(5.3)

If (X, Y) has the probability mass function (3.5), i.e., s = - 1, then the marginal probability mass functions of X and Y are, respectively, h_(.Iu,b+c,n)andh_(.Ib,a+c,n)asgivenby(1.6).Itiseasilyverified that in this case (5.2) and (5.3) again hold. Model ZZ. If (X, Y) has the probability mass function (3.4), then the marginal probability mass functions of X and Y are, respectively, h(.lu,, +u,,,u,, +a,, n) and h(.lu,, +a,,, ~,,+a,, n) given by (1.4). To compute E( Y I X=x) let U, V, and W be, respectively, the number of balls drawn with labels (1, 0), (0, l), and (1, 1) in the first n trials. Then X= U+ W, Y= V+ W, and E(V1 U+ W=x) is the expected number of (0, 1 )-labeled balls drawn on the (n - x) trials which yield (1,0) or (1, 1) and E( WI U + W= x) is the expected number of (1, 1 )-labeled balls drawn in the x trials which yield labels (1,0) or (1, 1). Thus E( V+ WI U+ W=x) is given by E(YIX=x)=(n-x)A+xA a01 + %o =

~~o,(~,,

a10

+~,o)+x(~,,~,--a,o~o,) (all+

683/35/l-S

a11 +

U,o)(Qo,

(5.4) + %o)

.

64

MARSHALL

AND

OLKIN

Here Corr(X, Y) =

~11%

-

~10~01

C(Ul,+ ~,a)(~,,+ ~,a)(~,,+ ao,)(a,o+ %dl””

(5.5)

When s = - 1 so that (X, Y) has the probability mass function (3.7), the marginal probability mass function of X and Y are h- (. 1a,, + a,,, a,,+u,,n) and h~(~~u,,+u,,,u,,+u,,n) given by (1.6). In this case (5.4) and (5.5) remain valid. Model III. When (X, Y) has the probability mass function (3.9), then X and Y have the marginal probability mass functions h( . ( a, b, n,) and h(.lu, 6, n2). Let U, W, and V be, respectively, the number of red balls drawn in the first n, -k trials, the next k trials, and in trials n, + 1 to n, + n, - k. Then X= U+ W, Y= V+ W, and E(YIX=x)=E(UIV+ W=x)+ E( WI I’+ W= x) and, from previously used arguments, it follows that

= h-kb u+b+n,

+xk(a+b)+v, n,(u+b+n,)

’

(5.6)

Here Corr( X, Y) =

k(u+b)+n,n, [n,n,(u+b+n,)(u+b+nz)]“*’

For this model the case s = - 1 is slightly different. The probability function (3.11) has marginals h _ (. 1a, b, n, ) and h (. 1a, b, n,),

(5.7) mass

(5.8) and Corr( X, Y) =

k(u + 6) -n1112 [n,n,(u+b-n,)(u+b-n,)]“*’

Notice that of all these correlations, only that of (5.7) is necessarily positive. This in turn is due to the fact that distributions of the form (3.6) are associated, and so the same is true of (3.9).

65

BIVARIATE DISTRIBUTIONS FROM URN MODELS REFERENCES

G. E., AND NEYMAN, J. (1952). Contributions to the theory of accident proneness, I, II. Univ. of California Publicafions in Statistics, Vol. I, pp. 215-276. Univ. of California, Berkeley. [Z] BOSCH, A. J. (1963). The Polya distribution. Statist. Neerlandica 17 201-213. [3] BOSWFLL, M. T., AND PATIL, G. P. (1970). Chance mechanisms generating the negative binomial distributions. In Random Counts in Physical Science, Geo Science, and Business, Vol. 3 (G. P. Patil, Ed.), pp. 3322. Pennsylvania State Univ. Press, University Park, PA. [4] BRICAS, M. A. (1948). Le q&me de courbes de Pearson et le schPma d’urne de Pdlya. Gauthiers-Villars, Paris. [5] DYCZKA, W. (1973). On the multidimensional Polya distribution. Ann. Sot. Math. [l]

BATES,

[6]

EGGENBERGER,

Polon.

Ser. I 11 43-63.

Z. Angew. [7]

F.,

Math.

EGGENBERGER,

AND

Mech. F.,

frequence. Comptes

AND

P~LYA, G. (1923). Uber die Statistik verketer Vorgiinge. 1 279-289. P~LYA, G. (1928). Sur l’interpretation de certaines courbes de

Rendus 187 870-872. An Introduction to Probability

[15]

Theory and Its Applications, Vol. I, 3rd ed. Wiley, New York. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. Wiley, New York. HALD. A. (1960). The compound hypergeometric distribution and a system of single sampling inspection plans based on prior distributions and costs. Technometrics 2 275. HEATH, D., AND SUDDERTH, W. (1976). DeFinetti’s theorem on exchangeable variables. Amer. Statist. 30 188. JANARDAN, K. G.. AND PATIL, G. P. (1970). On the multivariate Polya distributions: A mode1 of contagion for data with multiple counts. In Random Counts in Physical Science, Geo Science and Business, Vol. 3 (G. P. Patil, Ed.), pp. 143-161. Pennsylvania State Univ. Press, University Park, PA. JOHNSON, N. L., AND KOTZ, S. (1969). Discrete Distributions. Houghton-Mifllin, Boston. JOHNSON, N. L., AND Korz, S. (1977). Urn Models and Their Application. Wiley, New York. KAISER, H. F.. AND STEFANSKY. W. (1972). A Polya distribution for teaching. Amer.

[ 161

LUNDBERG,

[17]

MARSHALL, A. W., AND OLKIN, I. (1985). A family of bivariate distributions generated by the bivariate Bernoulli distribution. J. Amer. Statist. Assoc. 80 332-338. STEYN, H. S. (1957). On discrete multivariate probability function. Kon. Nederl.

[S]

[9] [lo] [ 111

1123

[13] 1141

FELLER,

W.

(1968).

Statist.

[lS]

26 4CH3. 0. (1964). On Random Processes and Accident Statistics. Almqvist & Wiksells, Uppsala.

Wetensch.

Proc.

Ser. A 54 23-30.

Their

Application

to Sickness

and