On the multidimensional Polya distribution

A N N A L E S S O C I E T A T I S M A T H E M A T I C A E P O L O N A E S e r i e s I : C O M M E N T A T I O N E S M A T H E M A T I C A E X V I I ( 1 9 7 3 )

R O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )

Wa ç l a w Dy c z k a (L ô d z )

On the multidimensional Polya distribution

0. Introduction. The characteristic function is not an efficient device for finding moments of the multidimensional Polya distribution. The purpose of this paper is to evaluate the moments of the multidimensional Polya distribution and to detect by these moments some properties of this distribution. In section one the ^-dimensional urn scheme of Polya -and distribution (1.1) connected with it have been defined. In section two this distribution has been generalized in the sense that p{ are arbitrary number from the interval (⁰.¹ ); this generalization is distribution (².²) which is the subject of the further considerations, its special cases are:

distribution (1.1), the multinomial distribution (2.5) and the Tc-dimension­

al hypergeometrical distribution (2.6). The factorial and ordinary moment of distribution (².²) and those of its special cases are considered in sections three and four; the recurrence formula for ordinary moments has been obtained in section five. In the next section it has been shown that dis­

tribution (².²) may be treated for a > ⁰ as the mixed multinomial and Dirichlet distribution. This fact has been employed to a repeated deduction of the moments of distribution (².²) from the corresponding moments of the multinomial distribution. The deduction of the ordinary and fac­

torial moments turned out to be simpler whiles in the deduction of the recurrence formula the calculations are also cubersome. Because of restric­

tions imposed on parameter a of distribution (².²) the formulas obtained by this method are not equivalent to those found earlier which are more general. In section seven we investigate the further properties of distri­

bution (².²) and of its special cases; related to the marginal distributions of this distribution, the conditional distributions, the distribution of the sum of random variables which are the components of a &-dimensional random variable with distribution (².²), and of the distribution of random vector whose components are disjoint sums of successive components with this distribution. Finally it has been shown in the last section that the Polya distribution is a non-singular distribution.

44 W. D y c z k a

1. The ^-dimensional Polya distribution connected with the /i diinen sional urn scheme of Polya. The one-dimensional distribution connected with Polya urn scheme may be generalized for the ^-dimensional case in the following way. Let a set of N elements among which there are Npi elements of the class C{, i = 1 , 2 , . .., & + 1, be given. Let, moreover,

fc+i

Ct be disjoint sets, hence Pi = 1. We choose at random n elements

i= l

from the set of all the elements according to the following scheme. We take at first one element and then put it back adding s ^ 0 further ele­

ments of class from which the first element has been chosen. We repeat that procedure w-times.' The result of n random choices is a sequence of n terms Ci among which there are x{ symbols Ct . Every such sequence has the same probability as the event consisting in the successive choices of : first œ1 elements of the class Cx, then x2 elements of the class C2, ..., and finally xk+1 elements of the class Ck+1. It is namely equal to:

fc+i

f ] (.Npi)lxi’- S]/ N [n’- S],

i= 1

where x[r,h] is a factorial polynomial with respect to x i.e. = x(x — h)...

... {x — (r — l)h) (moreover, we shall write x[r] instead of xlr>1^ if h = 1).

The number of all the sequences of n terms Ci among which we have xx, x2, . . . , xk+1 symbols CX, C 2, ..., Ck+1, respectively, equals the number of permutations of w-elements in which some elements are repeated xx, x2, ... , xk+1 times, respectively. Thus denoting by X xi ..., X fc+1,

к

where X k+x —n — ^ X {, the random variables whose values are the

i= 1

numbers of elements chosen from the classes Cx, .. . , Ck+X, respectively, we obtain the following probability distribution p ( x x, ..., xk) of the fc-dimensional random variable ( Xx, . . . , X k):

. k +1

(1.1) р( хг , xk) = »!

г = 1

If s < 0 we assume that

(1.2) ^ ( - s K m i n ^ , . . . , N pk+1).

If the last condition is satisfied (it is no restriction for n, if s > 0), then each x{ may accept any of the integer values from 0 to n, obviously with xx-\-... -j-£cft+1 = n. The random choice scheme described above may be called the k-dimensional urn scheme of Polya. In paper the probability distribution given by formula (1.1) shall be called the k-dimensional Polya distribution connected with the k-dimensional urn scheme of Polya and denoted by p( xx, . . . , x k) or by p { xx, ..., xk-, n; p x, ..., p k; s; N) if we wish to point out its parameters. Special cases of this distribution

Multidimensional Polya distribution 45

are: the multinomial distribution (for s = ⁰) and the /«-dimensional hyper- geometrical distribution (for s = —¹).

2. The /« dimensional Polya distribution. Now we shall generalize formula (see [²])

(x-\-yfr,h]

for j ^ ² components.

b ] y [ r - i , h ]

Le m m a 1 .1 . Let Kjn be the class of all systems (xlf ...,xf) of j non- negative integers such that x1 + ... -\-Xj — n\ let h be an arbitrary real number.

With these denotations the formula

(2.1) ( J^{» 1” ’ * 1} - и! V [ j («h"1/*.-!)

г^{= 1} Kj>ni=l

is true. The summation running through all the elements of the class K j n.

Proof. An elementary identity in t

j j

/ 7 ( i +«) ' i ==( i + t ) â , Ci

г = l

is for |/| < ¹ equivalent to the power series identity

Carrying out the multiplication on the left-hand side we obtain

Comparing the coefficients at tn we have

or

( У > , . р = „ ! у / / ¹¹ i=i Kjf1l i = ¹

After substituting in the last equality et = aj h for h Ф ⁰ and taking into account that

and multiplying both sides by hn we obtain the required formula. Formula (².¹ ) as the Newton polynomial formula is also true for h ⁰.

46 W . D y c z k a

Dividing the numerator and the denominator on the right-hand side of formula (1.1) by N n and putting a = s/N we obtain that formula in the form

*+i

(2.2) p ( xx, . . . , æk) = n l ] J (р ^ ’- а^ г.!)/1[,г’- а]

i= l

and condition (1.2) becomes

(2.3) й ( - а К ю т ( ^ , . . . , ^ +1).

In formula (2.2) the constants p t and a are rational numbers. We shall prove that this is a ft-dimensional probability distribution with arbitrary eonstans p { satisfying the obvious conditions:

fc+i

(2.4) P i >0, i = l , 2 , . . . , f c + l;- ^ Pi = 1

i= 1

and with arbitrary d satisfying condition (2.3).

Indeed applying formula (2.1) we obtian Jc+l

p{xx, . . . , x k) = (n]]±[tl,~a]) j j (PiXi~a] fal)

^k+l,n К-к+1,п^=^

k+1 г= 1

The condition p (x11 . . . , xk) ^ 0 must be proved only for a < 0. In this case (2.3) holds from which it follows that all the factors of the numerator and the denominator are positive.

The distribution given by formula (2.2) in which p { and are arbi­

trary real numbers satisfying conditions (2.3) and (2.4) shall be called the k-dimensional Polya distribution which wre write in the above for or, if we wish to point out the parameters, in the form p (хг , ..., xk; n ; p x,...

.. . , p k\ а; 1). The special cases of the Polya distribution (2.2) are: the

^-dimensional hypergeometrical distribution (when a — —1/ N and p t

= XPilN)

k+1

(2.5) **) = »! J ] ( ( Npj Wf al j / XM

г=1

and the multinomial distribution (a = 0) k+1

(2.6) p ( x1, . . . , æ k) = n ! J J (plijXil).

г = 1

Distributions (2.5) and (2.6) will be denoted by p ( x x, . . . , x k-,n;

P i , --чРк! N ) and p( xx, n \ p x, . . . , p k), respectively.

Multidimensional Polya distribution ⁴⁷

3. The factorial moments of the multidimensional Polya distribution * Corollaries. Let K j n stand for the sequence described before..

Th e o r e m 3.1. The factorial moment ^{a [ri}....rfc] of order г г -\- ...~\-rk

= r of the h-dimensional Polya distribution (2.2) is given by the following formula :

к

(3.1) aIrj....г*, = п П[ ] р Г« - ° > / 1 Ь- ° К

i = l

Proof. By the definition of the r-th factorial moment of a &-dimen- sional discrete distribution and by formula (².²) we have

к fc+1

(a) «[tl.... , * , = « ! У У ^ > / 7

K k + l , n V:=1 i=1

Let K k+l n c: K k+l n be the class of all systems (aq, ..., a?ft+1) of non-negative integers satisfying the conditions: + ...-\-xk+1 — n\

i = 1 , 2 , . The summation in the class K k+l>n, in equality (a) may be replaced by summation in the class K k+i>n because of Xi = ⁰ for xi < r i . If, moreover, we take into consideration the following equalities resulting from the definition of factorial polynomial

р\ч,а] = p P ’ a}(Pi + ria)lx* ri’ e]; xl{i]Iх,1.= Ц(х{ — г{)\-,

then equality (a) becomes (Ь) a[rlt....r*]

Jc Jc

n[r] J/ р 1^ ’~а] (n — r)l f ] (Pi + ri a)[xi~ri’~a]р [*ь+1,-al

___________i = l ______ i = l ___ __

- ¹^-«¹(¹ + г а )[«-л-«] 2 j

К_{k+ \,n} ^{] ] { X} i - r i ) \ x k +1^!

i = l

Let х{ — г{ = m{, i = l , 2 , . . . , f c ; xk+1 = mk+1; condition aq + ... + xk+ г

= n becomes now тг + ... ~hmk+1 = n — r and the class K k+1>n will be transformed into the class K k+l n_r of all systems (m1? ..., mk+1) of non-negative integers satisfying the condition + = n — r.

With these denotations the last equality becomes

(c)

n[r] ^{/ 7} р ^ a]

i=l ( n - r ) l j ] (р{ + г{а)[т" а]р {^ '+1’ a]

l [r’- a]( l - f ra][п — r, —a]

Кk+\,n—r

k + l

/ 7 * М

48 W. D y c z k a

The sum on the right-hand side of the last equality is by (2.1) equal to

( J ^ (Pi + ri<*)+Vk+ 1) [ n _ r ’ _ a l = ( 1 + r a ) [w- r' - aJ,

t=i

which after substituting into (c) gives required formula.

Corollary 3.1. The factorial moments a[ri for: the k-dimension­

al Polya distribution (1.1) connected with the k-dimensional urn scheme of Polya, the k-dimensional hyper geometrical distribution (2.5) and the multi­

nomial distribution (2.6) are given by the formulas:

к

<3.2) «■[,,...,*i = » I r , / 7

i= 1 (3.3)

к

<*(,,...rt ] = f ]

i=l

‘(3.4) % ...•*] = nMf ] P <‘ ’ г= 1

respectively. Formulas (3.3) and (3.4) are known (see e.g. [3]).

Proof. Formula (3.2) is identical with formula (3.1) with a = sjN and Pi = NpilN. Similary accepting s = —1 in (3.2) we obtain (3.3).

To obtain (3.4) we have to put a — 0 in (3.1).

Coeollary 3.2. For the factorial moments of order r of the k-dimen­

sional Polya distribution (2.2) the following recurrence formula

<3.5) a[r1,...,ri0_ 1,r<0.r<0 + 1,...,rft]

_ ( n - r + l ) ( pi0 + (ri0- l)a) l + (r _ l ) a at o - holds.

Proof. This formula is an immediate consequence of formula (3.1) and the definition of a factorial polynomial.

In the considered special cases formula (3.5) becomes:

(3.6) ( n - r + 1) (Npi0 + (ri0 - 1 )s)

a [rv . . . , r io, . . . , r k ] N + ( r - l ) s a [r i,. . . , r io- l , . . . , r k ]

for the ^-dimensional Polya distribution (1.1) connected with the k-di­

mensional urn scheme of Polya, (3.7 а[П....riQ’- ’rk\

{ n - r + l ) ( Npi0- r i0-hl)

N — r + 1 ^Ь...,Гк]

for the ^-dimensional hypergeometrical distribution (2.5), and

“['l. — ....rk \ ~ — r J r l ) P i 0 a [rl , . . . , r i o _ 1, . . . , r k ]

for the multinomial distribution (2.6).

Multidimensional Pôlya distribution 49

4. The ordinary moments of a multinomial Polya distribution.

Corollaries. Let 8] be Stirling numbers of the second kind i.e. the coef-

г ч

ficients in the identity: xri = £ S^.x^K тг=о 1

Th e o r e m 4 .1 . An ordinary moment ari> of order rx+ . . . + rk

= r of the Tc-dimensional Polya distribution (2.2) is given by the following formula

r k

(4.1) r V > r k

=

-

t 2к mi

J m v , —a ] /1 i = l ^{- « 1}

'A

m k=

Proof. According to the definition of an ordinary moment of order r of a ^-dimensional discrete distribution p( xx, ..., xk)

<4- =

х\1 . .-ХГ ккр{Х !, ..

К-к+1,п

since

к к Ч * ■ 1 гк

IJxï н

А - 2 «а. • .. 4W fcl

г=1 г = 1 тг-=0 т^=0 тк=0

thus гк

a rv ...,r k

= А _-

«а ,- % ^£ .. • •, ® fc) •

77^ = 0 _тк~° •^fc+l,7i

By the definition of a factorial moment of order r of a ^-dimensional discrete distribution we further have

r k

(4.2) h-i,—,rk ~ ••• ®mka[m... mk]

mj = ⁰ mk=0

If we take into account (3.1) in the last formula, we obtain finally (4.1).

Co r o l l a r y 4.1. The ordinary moments of order r for the Tc-dimensional distributions: (1.1), (2.5) and (2.6) are given by the following formulas, respectively :

r k [ SmAjLL д "4i " ^{[ 2к} m%3—SJ, (4.3) ^{ari Гк} = У ... J J ( N P i r ^ l N <-1 ),

m^{1 = 0} m^{^ = 0 * = 1}

ri rk [ 2 тг-] k [ ² »»t]

(4-4) ari у ... у ^ Ц ),

M l A m i - — A 1* — 1

m^{x= 0 mk = 0}

r l r k J 2) 7/1 ■] ^

o - 6) s ....-, = A - A «a, ’ П î-H m^{1 = 0} wi^{£ =0}

4 — R o c z n i k i P T M — P r a c e M a t e m a t y c z n e X V I I

50 W. D y c z k a

5. The recurrence formula for the ordinary moments of multidimension­

al Polya distribution. Corollaries.

Theorem 5.1. For the ordinary moments of the h-dimensional Polya distribution (2.2) the following formula

(5.1) a r i , . . . , r j _ 1,rj + l,rj + 1, . . . >rjc f ⁾ Pi ^{) * • * )} Pj-1) Pj “b Pj+1) ‘ ‘ ‘ ) Pk ) ® j f “Ь ®) “b (Pj/&■) Ap- ®rx,.. .,rj,...,rk

holds, where

‘fj, ~,r~ ,rk (Щ Р17 • • • ? P j) • • • ) P k ) a ) f )

and Ap.ar^ >r.t r]is the first finite difference with the step а Ф 0 of the moment

a rh ...,rjy ...,rk in relation to p j .

Proof. In the proof we will use of formula (4.1) which expresses the ordinary moments in terms of factorial moments. According to this formula

(a) arb . ..,rj+ \,...,rk

rX rj+1 _rk

= У ... V ...

mx — 0 шу=0 mk=0

S rkm k

[ 2 т {] к

■ <=1 П Pimp' al

г=1 [ 2 т{,-а ] к 1 М

For the Stirling numbers of the second kind appearing here the formula

S ri +1

holds. By this recurrence relationship we may divide the right-hand side of equality (a) into two terms

where

ri A = Z

m x=o

ri m1= 0

fh ‘x, . . . , r j + l , . . . srk A -f-

r j+1 rk n i =1 / 7 p l mi’~ a]

■ Z • • • 2 *3, •.. - Щ s rÂj-+\ - S ' A --- - ---

nij= 0 mk = 0

rj+1 rk П

■ 2 - Z 8r+ - 8\

nij=Q mk ~ 0

к f

l 2 mi t-a] i M к [ 2 т {] к

*=1 r j p ïm i,~ a]

г=1 [ 2 ггц, -а]

I г = 1

Multidimensional Pôlya distribution 51

We shall prove that В is equal to the first and A to the second term in formula (5.1). Taking out npi before the summation sign, because of 8%. = ⁰ for mi < ⁰, we have

4 r j + l r k

B = n Pj V ••• 2 ■■■

m j = 0 m , j = l m ^ . = 0

к [ - 1 + 2 nij] к

( n -¹) ^{г‘= 1} П Pimi’~a](Pj+

i = l

[ —1+ 2 mjt~a]

(l + a) ^{i = 1} Let nij — 1 = m'j, then

В = npj_mj

2

= 0

ri

m '. = 02 1

Гк 2 < b

m k =0

^ 7 Л ^ ^ ; л - . ^ x

X

, к [nij+ 2 mj)

( п -1) +

i = l

_____________ Фз__________________

(1 + a)

[wi,-+ 2 w,-,-a]

1 i=l J

ФЗ

By formula (4.1) we conclude that

В = npjarif гГк(п — 1 ‘, p lf . . . j P j - u P j + ajPj+i, . ..,Pki a; l + a).

Calculate now the first finite difference with the step а Ф ⁰ in relation to pj of the moment aTi....Г}_ given by formula (4.1). We have

^ O r I t . . .,r * = (< * lp -j)A

since

AlfPp ^ = (а/p,)mjPp ’-* ’ and 8Ц+1 = ⁰, hence

A — (Pj/a) Apj<xr^ r^.

Thus the theorem has been proved.

Co r o l l a r y 5 .1. For the ordinary moments of the multinomial distri­

bution (².⁶) the following recurrence formula is true

( 5 - 2 ) a r1,...,ri _ 1,r/+ l , . . . , r fc

d .

npjar^ r, ^r^{n 1⁵ jpl.? • • •, Pj, • • • » Pk) + Pi arlt...,rk, wJiere an....rk = «rx.... rk(n-, Pi, •••,Pk)•

52 W . T>yе л к а

Proof. The limit of the right-hand side of formula (5.1) as a -> 0 (distribution (2.2) tends then to distribution (2.5)) is the right-hand side expression of formula (5.2). In fact, by (5.1), (4.1) and (4.5) we obtain successively :

lim ^npj arit^mtrk(n — l ] p t , ■ - ■, р ^ г, p j + a , pj+1, . . . , p k; a; l + a) +

rk к

[ S mA

+ = npjlim £ ^ x

m- ==

m1=0 тд.=0 (Pi + a)[n4’- a]f l р Р ’~а]

i ~ l

X--- P --- + P j\im a-lAaPjari...rfc

[ 2 т г-, —a] a->0 i —1

к к

[ 2 Щ] * [ 2 mA

(1 + a)

ri rk

= n P j % . . .

2

/7

mx= 0 m k~° г=1

d d

"*■ ^ Qp ■ ari’-"’rk ~ nPiar1,...,rk(n — Pi» •••»P*) + dp. —.rA*

The last two members of this sequence of equalities contain the mements of multinomial distribution (2.5).

Substituting in formula (5.1) the appropriate values for the para- meters a and p { we may obtain the recurrence formulas for distributions (2.5) and (1.1).

6. The multidimensional Polya distribution as mixture of distribution.

Application. Is is known (see [1]) that a one-dimensional Polya distribution may be treated as the mixture of a binomial distribution in relation to the parameter p and the beta distribution (with suitably chosen para­

meters). This property shall now be generalized to the Tc-dimensional case and employed for obtaining in another way the moments of Polya distribution from the corresponding moments of the multinomial distri­

bution.

A ^-dimensional distribution with the density

(6.1) / ( $ 1 , . . . , ‘&k ‘, (Oj, . . . , COk )

Г(со1 + ... + Qfc+1) ^ - 1

= r (oii) ... r(cok+1)

0

t 1 f o r , . . . , # k ) c S k ,

for (0J, ..., &k)4Sk, is called (see [3]) the Birichlet distribution, where Sk is a simplex in the

^-dimensional euclidean space defined by the inequalities : 0

к

(г = 1 , 2 , . . . , Тс), «! + ... + ©*< 1; vk+1 = 1 Vf, со* (г = 1 , 2 , .. . , &+1) г = 1

Multidimensional Polya distribution 53

are arbitrary numbers. Let the parameters of a Dirichlet distribution be of the form: = Pil° (i = 1, 2, ..., ifc + 1), where a > 0, P i > 0 (i = ¹ ,² , . . . , Tc), pk+1 = ¹ — (Pi + --.+Pft); then density (⁶.¹ ) becomes (⁶.¹a) /(# !, &k',Pila, . . . , p kja)

= (Г(11а)1Г(Р11а) . . . Г ( р к+11а))П'1а-' ... 0 f f i^{l / e - 1} or 0 for ( ^ , ..., 0Л) e and (^i, .. . , &k) 4 Sk, respectively.

Let, moreover, multinomial distribution fe+i

(⁶.²a) = nl [ ] (0р/ х{1)

i—l

be given, where x{ (i = ¹ ,² , ..., 7c+ ¹ ) are non-negative integers, xk+1

= n — (xx + •.. +^fe) -, 0 {i — ¹ ,² , ..., к + ¹), '&k+x ^{= 1} i'&x + . . . + vk)-

Th e o r e m 6.¹. The Jc-dimensional Polya distribution (2.2) with a > 0 is the mixture of the multinomial distribution (⁶.²a) in relation to the para­

meters vx, . . . , v k with the Dirichlet distribution (6.1a).

Proof. The probability that a random variable with conditional multinomial distribution (⁶.²a) takes the admissible value (xx, . . . , x k) is equal to

Pr(Xi = x1, . . . , X k = xk) i i

= / • • • f p ( x i , •••> %k', n ;d x, ..., dk)f(êx, ..., dk-,px/a, . . . , ^ / a ) ^ ... dêk

о 0

if the parameters (#x, . .., êk) of this distribution be treated as a reali­

zation of the random variable ( dx, ... , 0k) with the Dirichlet distribution (6.1a). Now expressing the Dirichlet integral in terms of the gamma func­

tion and making use of the properties of that function we obtain succes­

sively :

Pr ( Xx = xx, ..., X k — xk)

k +1 1 1 * + l

= (n '.r ( lla )l[ J x( i r ( p f/a))J . . . j f ] M, ... dÿk

i=l 0 0 /=1

A —f—1 1c~\~ 1

= ( n i r ( l / a ) / ] J х{1Г(р{/а)}

■07

i—l i—l

k+1

= » ! / 7 i=l

k+1

= n\ f j ( p ^ ’~a] /х{!)/11п’~а] = p ( x 1} . . . , x k; п ; р и jbfcj a; ¹)

54 W. D y c z k a

which is the probability of the fact that the random variable (X x, ..., X k) with the k-dimensional Polya distribution (2.2) takes the value (xlf .. . , xk) if the of values of the parameter a be confined to the positive numbers.

An immediate consequence of this theorem is the following Theorem 6.2. The ordinary factorial moment of order r of the k-di- mensional Polya distribution (2.2) with a > 0 is the mean value of the r-th factorial ordinary moment of multinomial distribution (6.2a) if the parame­

ters (#!, .. . , fyf) appearing in the moments of that multinomial distribution be treated as a realization of the random variable (6X, ..., 6k) with the Dirichlet distribution (6.1a).

Applying three times the last theorem we may obtain in a shorter way formulas (3.1), (4.1) and (5.1) concerning the k-dimensional Polya distribution (2.2) with a > 0 from the moments of multinomial distri­

bution (6.2a) given by simpler formulas (3.4), (4.5) and (5.2), obtained earlier as special cases of the moments (3.1), (4.1) and (5.1), respectively.

In the sequel we denote by а[Гъ>Гк], ar.t (if no special definition is given) the moments of the k-dimensional Polya distribution.

(i) In accordance with Theorem 6.2 and formula (3.4)

« h ....* = е(и1г№ ) = иМЕ№ ) -

г=1 г= 1

Е (П Qrf)

i= k

(see [3]) we have

к к к

Е( Л »?) = ( 1 7 <ft/“)[ri,_11) / ( l / a p ' - 11

г=1 *=1 г=1

and consequently

«Pi....* “ K T I r i ' ' - - 1) / 1''’- 1

t =l

which is formula (3.1) for a > 0.

(ii) Similary one can obtain formula (4.1) from (4.5) and Theorem 6.2:

rk

%.... , = S -

mx—0 mJc~ 0 ^{г = 1}

__ rk , Г 2 шА k f 2 те,-,— a] ,

m1=0 mk=0 i= 1

which is formula (4.1) for a > 0.

Multidimensional Polya distribution 55 (iii) In the two cases considered so far one could make use of the known formula for the r-th moment of the Dirichlet distribution. To obtain formula (5.1) it is necessary to carry out the immediate calcula­

tions. According to Theorem 6.2 and formula (5.2)

( 6 . 2 ) a r1,...,rj _ 1,rj + l ,r j +1,...,rk

= E (ndjCl^ г. )Гк(П’, дг, . . . , вк)) + E a rx,. ...,rkj

the moments under the symbol of the mean value are (only here) the moments of multinomial distribution (6.2a). The further calculations con­

sist of making use of formula (4.5), the relationship between the Dirichlet integral and the gamma function and of a recurrence relationship for the gamma function :

, пГ(Ца)

-A- — fc+i x

П r (Pila)

i= l

1 l rl rk

X

к 1.2 mi ]

I ■■■/2 ■■■ 2 82 ‘ ■■■

о о ^{m± = 0 m k —°}

x J I ...d& k , ^1,1

i —\

ФЗ П ПрМ >«¹⁼⁰ o

i —1

r * Г ( Р з 1 а + ш ^ 1 ) г ( р к + 1 / а ) / 7 r i P i l a + M i

x ( » _ l / A ^{” 4'1}---i p t i --- Г(¹ /а + .¹ + £ m{)

i= 1

П

rl rk . к

\ -i ^{[ 2} т{\ I

2j --- 2 j «Й, l(î-,/a)I”i +1- 4 X ma=о m^=o

к к r \ rk

Я

î = l,+j Ш| = ⁰ тк^{= 0}

к к к

[ .2 mil [ | ж 1.2 т{,-а]

X ( W - 1 ) ,=1 (p,. + a )[mi ’- a] I l plmi’~a]/( 1 + а ) г=1

V г — 1,ФЗ

= n P j a r1,..., rk ( n - l ; P i , . . - , p j - i , . P i + a , p j + 1 , . . . , p k -, а ; 1 + а ) ;

56 W. D y c z k a

в = E l в.

Г1 Гъ

4 . 1 - 2

к к

[ '2 т^] ^{— у}

Цаг*

Г( 1/а

т х — О тк =0

1 1 Г1

г=1

С Г VI \ 7 r . i ”4]к

j . . . j 2 - 2 % ^ i- « v =1 w -1 х Aj-J- 1

П r{Pil°) 0 0 ml = ° mk^°

i— 1

Г (1 /а ) »•! rk

n J-

ё щ1«+т{- 1 ^ . . . dtf* = *+ï ^

i=1 П r(Vila) mx = O m^O

X У! ...

x wl2mi]r ( Pk+l l a) nr (Pila + mi) Д. JÜ

г=1 *=l

1.2к ™il Г(1/а +

г = 1

И12 = 0 Шд.= О

X

/ |—г fC [ 2 т^,—а]

г=Г

(PjI® ) .... r/X

By J. and В relationship (6.2) coincides with recurrence formula (5.1) if а > 0 in (2.2). The formulas obtained in this section are not equivalent to formulas (3.1), (4.1) and (5.1), because of the larger range of variability of a.,

7. Some properties of the multidimensional Polya distribution. Ap­

plication of factorial moments. Corollaries. The properties of the multi­

dimensional Polya distribution investigated here are similar to those of the Dirichlet distribution (see [3]).

Th eo rem 7.1. I f a joint random variable ( Xlf ..., X k) has the multi­

dimensional Polya distribution p (xx, .. . , xk ; n ; p x, ..., p k ; а ; 1) given by formula (2.2), then the marginal distribution of the random variable Xky), h < k, is the к x-dimensional Polya distribution p (xx, ..., xki ; n > Pi ? • • • » Pky 5 a ? 1 )•

Proof. Let in formula (3.1) kx < к and rki+1 = rki+2 = ... = rk — 0.

Thus we obtain the formula for the factorial moment of order rx + ... -f- rk of the marginal distribution of ( Xx, ..., X k ) in the Jc-dimensional Polya distribution (2.2):

(7.1) lhi ’■■■>rk11 n

ki k ki

[ S Г{] [ 2

i=1 / I ™[г; , -а] /1 г=1

f ] p l 'A i=l

This formula gives also the factorial moment of order rx - f ... + rk of the Aq-dimensional Polya distribution p (xx, ... , xk ; n ; p x, ... , p k ; a ; 1). But, on the other hand, a k-dimensional Polya distribution is uniquely deter­

mined by its moments, thus the theorem has been proved.

Multidimensional Polya distribution 57 Corollary 7.1. If a joint random variable { X x, . . . , X k): (i) has the k-dimensional Polya distribution connected with the k-dimensional urn scheme of Polya, i.e., the distribution p (xx, . . . , x k-, r; p x, ..., p k] s; N) given by (1.1), or (ii) it has the hypergeometrical k-dimensional distribution p (xx, . . . , xk ; n i Pn • • • j Vk'i N) given by formula (2.5), or (iii) it has the multinomial distribution p{ xx, ..., xk, n-,px, . . . , p k) given by formula (².⁶), then the marginal distribution of the random variable { X x, . . . , X kj), kx < k : (i) is the k-dimensional Polya distribution p( xx, . .., xk ; n ; p x, ..., p ki ; s ; N) connected with the кx-dimensional urn scheme of Polya, or (ii) it is the kx-di- mensional hypergeometrical distribution p (xx, ..., xk ; n ; p x, ..., p ki ; N), or (iii) it is the multinomial distribution p ( x x, ..., xk ; n\ p x, ..., p kf).

The validity of that corollary follows from the fact that the distribu­

tions in question are special cases of Polya distribution (2.2).

Theorem 7.2. If {Xx, . . . , X k) is a random variable with the k-dimensional Polya distribution given by formula (².²) and p{x\ n\ p) a; ¹ ) denotes a one-

dimensional Polya distribution given by the formida {n\ p[x,~a^ q[n~x,~ai j , then the conditional random variable (Xk | X x, . .., X k_x) has the one-di-

k- 1

mensional Polya distribution p [x ; n — £ x{; p k/(pk+ p k+x); aj(pk-\-pk+x)', l)»

i- 1

Proof. The distribution of the conditional random variable {Xk\ Xx, . . . , X k+x) shall be obtained by dividing the distribution of the {Xx, ..., X k) by the distribution of the random variable { Xx, ..., X k_x)y the latter being determined by Theorem 7.1:

p( xx, n \ p i , . . . , p k, a; ¹] p( xx, . . . , x k_x ; n-,px, . . . , p k_i; a; ¹ )

n\ t1 -

( n - 2 х{)1[ ] х<1 i— 1 i= 1

n:

к [n— 2 xj, — a] к

г= 1 l j p \ Xi ’ - a]

i= 1 i= 1

q[w,-a]

( 1 - I p

к— 1 [n— 2 Xj, — a] k—lk-l

1 Г 7

г = 1 f ] р\х°

i= 1 k—l k—l

\n - ¹ ><)! n xt \

i—1 г=1

Ïl Xi) p lk k,~a] (i - ^ Pi)" i=1 1 ' /(l - £ Pi) {=1

[ П — 2 X { , — a \ k-l k-l

[n — 2 X j , — a ]

k-l к k-l

, . I n - 2 X j , — a] [ n - 2 X ; , — a]

= ( Â Xi) p lk k’~a]Pk+li=1 liPk+Pk+l) <i=1

58 W. D y c z k a

If now the last member of the above sequence if equalities be divided by

к- 1

n — 2

(pk +Pk+i) *=1 ? we obtain ultimately

<7.3)

fc-i

n — 2 x .

i=iXi\(Pkl(Pk+Pk+i))lXk’ a,(Pk+Pk+l)] ^X xk '

к

[ n - 2 X i , - a / ( p k +Pk +i)] [n

x(Pk+il(Pk+Pk+i)) г=1

A

k-1

- 2 x i t - <tj{pk + P k + i ) ]

г=1

which is the one-dimensional Polya distribution of the required form.

Corollary 7.2a. I f a joint random variable { Xlf ..., X k) has the k-dimensional Polya distribution connected with the k-dimensional urn scheme of Polya, i.e., the distribution p ( x±, ..., xkm, n; р г, ..., p k; s; X) given by formula (1.1) and p(x \ n\ s; p\ X) denotes the one-dimensional Polya' distribution connected with the urn scheme of Polya, i.e., the distribution given by formula

{ l ) ( X p f ’~s]{ X ( l - p ) ) ln~x,~s]/ X ln,~s],

then the conditional random variable {Хк\ Х г, ... , Х к_г) has the one-di-

k- 1

mensional Poly a distribution p(xk' , n - x{; p k/{pk + p k+1); s/(pk + p k+1)', X)

г'=1

connected with the urn scheme of Polya.

Proof. It suffices to accept in (7.3) p t ~ Xp J X, i = k, k + 1, and a = sJX. Then after some simplifications distributions (7.3) becomes:

<7.4)

fc-i n — 2 X,

i = l г

xk

N V \lXk'~sliPk+pk+l)l

--- ^ — ) X

P k + P k + J

к к- 1

/ ЛГ™ ^-«/(Pfc+JPi+ï)] [ n - 2 х {, - а Ц р к + р к+ 1)]

I лРк+i j г=1 /N г=1

\P k+P k+

J

thus it is the Polya distribution connected with an one-dimensional urn scheme of Polya of the form announced in the corollary.

Corollary 7.2b. I f a joint random variable { Хг , ..., X k) has the k-dimensional hyper geometrical distribution р( хг, ... , xk ; n ; p x, ..., p k ; N) given by formula (2.5) and if p{x-, n-, p \ X) denotes the one-dimensional hyper geometrical distribution, then the conditional random variable { Xk\ Xx, . . . , X k_x) has the one-dimensional hyper geometrical distribution

k - l

p( xk; n - 2 1 XijpdiPk + Pk+i); Х ( Р к + р 1+к))-

г=1

Multidimensional Polya distribution 59

Proof. Accepting in (7.3) p { = Np^N, i = Те, h + l , a = —1/N (which means s = —1 in (7.4)) we obtain:

(7.5)

к - 1

n~ E Xi

г = 1

xk

' N pk ' Pk + Pk+h

[хк,ЩРк+Рк+1^

I x

X ' N pk+1 ’ Pk + Pk+¹,

к

[ n - E H M { P k + P k + l ) \ in

, ^{1 = 1} IN

k^{- 1}

E xi t l/(pk +pk+1)]

i—l

к- 1

- Уг^{= 1} г x k

{Npki x^( Npk+1)

к к — 1

[ n - 2 жг-] [n - E Щ\

( = ¹ //AT/~ I /(A(^fc + pfc+i))^,v> \\ ^{г' = 1} which is the hypergeometrical distribution with the announced para­

meters.

Co r o l l a r y 7.2c. I f ( X x , . . . , X k) is a joint random variable with the multinomail distribution p( xx, ..., xk, n m, p x, ..., p k) given by formula (².⁶) and if p(x-, n;p) denotes the binomial distribution, then the conditional random variable ( X k \ X 1, . . . , X k_ x) has the binomial distribution p(ook,

к- 1

2 ®i-,Pkl(Pk + Pk+1))-

г'= 1

Proof. Putting in (7.3) a ^{= 0} we obtain

■ S x{k-l

i —l 1 X,,

{Pkl(Pk+Pk+l))Xk(Pk+ll(Pk+Pk+l))

n — к2 Xj i — 1

this is the binomial distribution with the required parameters.

Th e o r e m 7.3. If ( Xx, . . . , X k) is a joint random variable with the h-dimensional Polya distribution p ( x x, nj p x, . . . , p km, a; ¹) given by formula (2.2) and p(x-,n-,p-, a; 1) denotes the one-dimensional Polya distri­

bution, then the random variable X — X x + ... -f X k has the one-dimensional Polya distribution p { x \ n ; p1-\-...-\-pk‘, a; 1).

Proof. The factorial moment of order rk+l of the random variable

к

Xk+1 = n — £ Xi is given by the formula

г=1

(7.6) E(Xjg+i]) = £ x[r^ p ]p ( x 1, . .., xk', n-,px, . . . , p k', a; ¹).

K k + l , n

After carrying out calculations analogous to those in the proof of Theorem 3.1 we obtain

(7.7)

The right-hand side of (7.7) is the factorial moment of order rk+1 of the one-dimensional random variable with the Polya distribution p(xk+1m, n ‘,Pk+i ; a ; l ) , i.e., p (xk+1; w; 1 - {px + ... + p k) ; a; l), thus Xk+1 as

60 W. D y c z k a

a bounded random variable has the Polya distribution p (%+1; n\ l — {px + . . . -\-рк) ’, a; l). Hence it follows that p(x; w ; Pi + . . . + р л; a; 1) is the distribution of the random variable X = Х г -f ... + X k.

Taking into account the fact that the Zc-dimensional distributions (1.1), (2.5) and (2.6) are special cases of distribution (2.2) we obtain

Corollary 7.3. I f { Xt , .. . , X k) is a joint random variable with a k-di- mensional distribution which is: (i) the Polya distribution connected with the Jc-dimensional urn scheme of Polya, i.e., the distribution р( х г, ..., xk}

n; Vii • • • ? Pk7 -$") given by formula (1.1), or (ii) the hyper geometrical distribution p( xx, ..., xk, n-, р г , ..., p k, N) given by formula (2.5), or (iii) the multinomial distribution p (хг, . . . , %; n; р г , ..., p k) given by formula (2.6); moreover, lei, p(x-, n; p; s; X), p{x\ n\ p; N) apid p(x',n-,p) denote the same as in Corollaries 7.2a-7.2c, then the random variable X = X x Y ...

. . . Y X k has a one-dimensional distribution which is, respectively, (i) the Polya distribution p (x; n-, p 2 -f- ... -\-pk; s; N) connected with the one-di­

mensional urn scheme of Polya, (ii) the hypergeometrical distribution p(x; n\

р г + ... Y p k', X), (iii) the binomial distribution p { x - n \ p1Y ... + p k).

Now we shal generalize Theorem 7.3. In this theorem we have con­

sidered an one-dimensional random variable which is the sum of all the componentes of a k-dimensional random vector with the Polya distribu­

tion ; this variable has also a Polya distribution. The question may be raised of whether the distribution of a joint random variable whose components are disjoint sums of successive componentes of a ^-dimensional random vector with a Polya distribution is also a multidimensional Polya dis­

tribution and, if so, in what form. The problem just formulated may be put more precisely in the following way.

Let the joint random variable X = ( Xx, .. . , X k) has the ^-dimensional distribution р( х г, .. . , xk-, n; p ±, ..., p km, a; 1) given by formula (2.2).

Consider the random variable Y = ( Y 1, .. . , Y{), where

к J A'j-t-k-2 k v \ Ay i

r , = 2 x >’ r « - 2 x » ~ : ' r < = 2 x r> £ * » < * •

j ~ I j —A*| + l j ~ k \ 4- ■ • • + k ^ i + l J —1

Let, moreover,

k \ к i Ay Ay \-. . . -(-Ay k-\ l

P i = ^ P j j P 2 = P j , - " 4 P i = £ Pj i Pu i = £

j = l j = k ^ r l j — j - ... i 1 j - - k \ -J-.,. - ) ■ -f- 1

With this denotations the following lemma is true

Lemma 7.1. If a random variable X — ( Хг , ... , X k) has the Polya distribution р( хг, . . . , %; n-, р г , ..., pfc; a; 1 ) given by formula (2.2), then the random variable ( Y x, и , . .., X k) has the (k — Tc1 Y l)-dimensional Polya distribution p (уг, xki,. t, ..., xk ; n ; pj, p kl f x, .. . , p k ; a ; 1).

Multidimensional Polya distribution 61 .

Proof. According to the definition of the mean value of a function of a random variable we have

(7.8) À Х >'И

3= kx+ 1

' к

= x'k^ tÛ ••• xlp ][ n - y 1- £ ц ) 1Г1]р(х!, . . ■, Хк; п; р1, . . . , р к;а-,1).

Kk + l,n J = k1 + l

By calculations analogous to those i the prove Theorem 3.1 we obtain (7.9) E « * J Î^:11 ... ( п- Y , - £ X,)™)

j = k x+ i

n

к к

[ r i + 2 rj] , * [r\+ 2 r j , - a ]

*“ *1+1

р[Г1\а] f ] Pp’~a]l

1

The right-hand side of (7.9) is the factorial moment of order rkl+i~ir • • • + rfc + ri °f the (fc—fcj + lJ-dimensional random variable {Xki+1, X k, X k+1), where Xk+1 = n — Y x — (X^{f c + 1} + ... + X fc), with Polya distribu­

tion of the form:

<7.10)

= nl ¹ — i—kx+i

= nl

t - i » • • • :» X k +1 > P k x+ l i “ ' 7 P k + 1 7 a 5 1 ) f t + i

2 ^_qJ k~ I

r /7 p p , ~ a]i i i n , ~ a

1 j —kx - \ - 1

.

j = kx + 1 ? =

Л + 1 k^{-\ - 1}

, ' [ 1 / 1 , - “ ]

[ J j . p > - — l / l l " ' - 0 l S r » ! Y x i

j = k x+ l j = k x+ l

Hence it follows that the random vector ( Y ly X k +1J X k) has the Polya distribution p [ y x, xki+x, ..., xk, n; p[, p h+1, . . . , p fc; a; ¹). Thus the lemma has been proved.

Le m m a 7.2. I f a random variable ( YX1 X k+x, ..., X k) has the Polya distribution p ( y lt xkl+1, . . . , xk] n; p kl+1, ..., p k] «; ¹) given by formula (7.10), then the random vector ( Y x, Y 2, X k +k +1, ..., X k) has the Polya distributionp{y^ y21xki+k2+1, ..., xk, n] p[, p'2, p ki+k2+1, . . . , p fc; a; ¹ ).

The proof is similar to that of the foregoing lemma, the only difference being that we start from the mean value

j=k^-f &^{2 + 1} and make use of (7.10).

Bepeating this argument we see that the following lemma is true

Le m m a 7.3. I f the random variable X — ( Xx, ..., X k) has the Polya distribution p( xx, ..., xk-, n; p x, ..., p k] a; ¹) given by formula (².²), then the