The multidimensional truncated Polya distribution

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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 L P O L O N A E Se rie 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 L X I X (1977) 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

Sé ria I : P R A C E M A T E M A T Y C Z N E X I X (1977)

Tadeusz Gerstenkorn (Eôdz)

О. Introduction. As the starting point we take the multidimensional Polya distribution given by

(0.1) p{æu æ2, 5 V l ’> P ii • • • 5 Pk'i -L> a) n\ ^{* + l}

ПН.— Л

PK-,-a]

i

?

where pi > 0 (i = 1, 2, &+1), *+iPi = 1, w is a fixed non-negative

i = l

integer, a is an arbitrary real number satisfying the condition n { - a ) = min(pa, p 2, . . . , p k+l)

(this condition can be missed if a > 0) and æ[r,h] = _ ( * _ ! ) & )

i = l

is a factorial polynomial of r-th order with respect to x (it is also called a generalized power of so) with stej> h (h — any real number).

Each of these variables of the distribution (0.1) may accept any of the integer values from 0 to n but in this way that the condition хг +

+a?2d- . . . -f% +1 = n is fulfilled.

We shall replace the notation p(soi,so2, ..., а?й; n; p lt p 2, ..., p k\ 1; a) of the distribution (0.1) by a shorter one p {xx, ..., œk).

This discussed probability distribution (0.1) can be connected with the so-called k-dimensional urn scheme of Polya described in [1], p. 44.

The information about the multidimensional Polya distribution one can find also in [2], p. 29.

The purpose of the paper is the establishment of the multidimen

sional truncated Polya distribution on the basis of the distribution (0.1).

The values of the random variables are truncated below fully arbitrary, i.e., the zero-truncated distribution is here only a special case of the above one. This approach to a subject increases the difficulty step of research

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but it gives simultaneously a wider range of using the results and proper

ties presented in the following sections.

In Section 2 the factorial moments of the discussed distribution are investigated in the case of truncation both for one and many variables.

The factorial moment of order r = r1Jr r2-\-... -\-rk of the fc-dimen- sional joint random variable (X xi X 2, ..., X k) with the discrete distri

bution p{xxi xk) is defined by

(0-2) a irx,...,rk ] =

x l xk

where x[[ i] (i = 1, 2, ..., k) is a factorial polynomial and the summation is running throughout all the values of the random variables X x, X 2, ..., X k.

In Section 3 the ordinary moments (moments about the origin) of the truncated distribution are discussed.

The ordinary moment of order r = rx -\-r2 + ... + r k of the fc-dimen- sional joint random variable (X x, X 2, . . X k) with the discrete distri

bution p {xx, x2, .. •, aok) is defined by

(0.3) arb...,rk = •••,«*)>

x i xk

where the summation is over all the values of the random variables X xi X 2, •.•, X k.

The results presented for the Polya distribution allow to obtain the analogous ones for the special cases of the discussed distribution, i.e., for the multinomial and hypergeometric truncated distributions.

In deducing of the theorems we make use of the Newton poly

nomial formula given in [1], p. 45, and cited here below.

Let Kjn be the class of all systems {xx, x2, ..., %) of j non-negative integers such that xx -\-x2 + . . . -j-Xj — n\ let h be an arbitrary real number.

Then with these notations the following formula (0.4) { £ b t} ™ = n ! £ П ( Ь ^ % 1 ,

t=l ^{Кл „}^{г = 1}

is true. The summation is running through all the elements of the class K.39П •

1. The multidimensional truncated Polya distribution. In the Polya distribution with the probability function (0.1) we restrict the range of variability of one of the random variables, e.g. X 3-, in this way that

c + 1 < Xj < n, where c is an integer. Because of the condition

xx -\-x2 + . . . + = n

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Multidimensional truncated P ôlya distribution 191

the left-hand side restriction by c of the vaines of the random variable X j involves the right-hand side restriction of other variables, i.e.

0 < xi ^ n — ( c + 1) for i = 1, 2, . j —1, j + 1 , . . . , & + ! .

For the given conditions imposed on the random variables we deduce now the probability function of the truncated distribution of the joint random variable (X x, X 2, ..., X k).

Let the class K k+l n have the meaning as given above in Section 0 (introduction). Let K k+ln denote the class of all systems (œ1} æ2, ..., ock+1) of к + 1 non-negative integers such that

0 < aSi < n — (c+1) for i = 1, 2, j — 1, j + 1 , &+1 and

#1 + • • • ~\~®к+1 = П‘

Let K k+ln be the class of all systems se21 ...,a?A+1) of &+1 non

negative integers such that

0 < Xj < c; 0 < < n for i = 1, 2, ..., j —1, j + 1, ...,k + 1 and

+ a?2 + •.. + tük+1 = n .

With regard to the above notation we can write the following relation

(1-1) K +1,n = K k+1>n\ K ''+1>n since

■ ^-k+l,n = -^fc+l,nu -^k+l.n K k+hnn K k+hn = 0 .

In order that p *(x 1, ..., œk) may be a probability function of the truncated Polya distribution it must satisfy the conditions:

(!-2) £ V * ^ x, .,.,œ k) = 1 and p*(co1, ..., cck) > 0.

к_k+l,n

After using formula (0.1) and relation (1.1) we have

^ P i æ l l • • • ? x k ) — ' У 1, P ( ® l i • • * ? ® k ) ^ P i ® l i • • * J M j . )

Kk+l,n Kk+l,n Kk+l,n

П!р1* ' - а]р1? - а>- Р % Г - а>

= 1

Kk+l,n2 l [n’ æ2\ ... a?fc+1!

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Determining the values of Xj successively 0 ,1 ,2 , .. . , c we obtain the systems of numbers

{ X \ , x2, . . X j _ n^{0 ,}X j + 1 ,^{. .} • ? ^ f c + l ) ?

( Х ц X 2^{, . .} ^{• ,}X j _ x , 1 , X j+ 1 , . .• ? ^>

( X i^,^x2, • • • 1 * * 7 - 1 7 c r x j + i ? ^{• •}• 7 ^ & + l ) •

The above systems satisfy the following conditions one after another

Xi⁺ x2⁺ ^{. . ■.} -\~Xj_x -\-Xj+x⁺ ^{. .} ^. ⁺ ^я ?л+1 ⁼ n

(1^.3⁾ ^X¹^-\-X²⁺ ^{• • .} ^-\-Xj_x⁺ ^{â f y}+1⁺ • • • + x k+\ = n

х г + X2⁺ ^{. .} ^• ^{Jr X j _}1J r X j + 1 Jr . .. + як+1 ⁼ n

Now we introduce into consideration the class K k>n_x., where Xj

— 0,1, ..., c, in order that there may be used the conditions given in assumptions for the variables X i and relations (1.3) which appear for these variables if the establishments with respect to values of X i are changing.

We shall understand by K k>n_x. the class of all systems {хг, х 2, ...

..., Xj_n Xj+i, ..., xk+1) of к non-negative integers which satisfy the appropriate condition (1.3) at the fixed value of Xj (Xj = 0 , 1 , ...,c ) .

Taking the above into account we can write

J ? P ( x n

^ k + l , n

— ¹

. . . , x k ) = 1 •

K- k+1,11 c [x.-, - a ]

V ^{P j}3 V ⁿ

, ^[Хл ,-a] ^[x2, - a ]

n \ p \ 1 p2 \Xk+l’ - a ]

• • • ^{P k}+1 1 [№’-°]^ 1! ^ 2!

! «1*1* “ “I ^{J X) ~}

• P i " ' P j -1

... ^xk + 1!

1, - ° } [ X j + l . - d ]

P j+1 ■ • • ^{P k + l}«[**+!■

x,j ! Z-j

Xj - ° К k,n — Xj

l[« ,-« l^{X i \} ... X j _ x \ x j + l \ ...

According to the definition of the factorial polynomial and formula (0.4) for the generalized Newton polynomial is

р ( х г , . . . , x k) ⁼ 1^- ^{^}

^k+l,n

с [хЛ [xu —a]

— П 3 p) 3

X

x у (»-% )'■ p ^ al ■ ■ ■ ^{P j - l} ^{P j +}¹ ■ ■ ■ p f t r - *

Kk,r vj — l • Mj + l ■ 7c+ 1 •

c [xj] [хл,—а

\ 1 71 P j [ n —х г , — a ]

= 1 ~ -]Sn’la^x-' ••• + P j- i+ P j+ i+ ••• + ^ +1)

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Since we can write

P 1 + P 2 +^{• • •} + P k+i ⁼ 1

(1.4) p (xi, ...» ®ft) = 1

С [ж,-,- a ]

у м v _ (1

\ xj) pin, - a] ^x

[n — xy,— a]

Following the definition of the marginal distribution of the random variable X i in the &-dimensional distribution of the joint random variable (Х г, X 2, ..., X k) it is easy to note that

(1.5) pAooj) P(®u œk) =

K-k,n—Xj

P[ X j , — a ]

Vi. [n—Xj,—a]

is the probability function of the one-dimensional marginal Polya distri

bution for 0 < CCj < n.

ЪеЬ Fj(a)j) be the distribution function of (1.5). According to the definition we have

(1.6) !V(e) =

X-- = 0 e_X*=0 c,

[ X j , — a ]

i i^ [ w , - a ] ____a ~ v ) ln~XiV х P j ) -a]

Using formula (1.6) we can write relation (1.4) in the form (1.7) J E p {xly ..., xk) = 1 - F j{ c ) .

K k+ l,n

We determine the probability function p*(xly ..., xk) by dividing the probability function in (1.7) by 1 — Fj(c) and we obtain

Th e o r e m 1.1. I f we are given probability function p(xly ..., xk) of the h-dimensional joint random variable (X ly ..., X k) of the Pôlya distri

bution (0.1) and the values of one of the variables, e.g. X j (1 < j < h), are truncated below by c, then the probability function p*(xly xk) of the obtained truncated distribution is determined by

(1^.8⁾ P*(fflи ‘ ‘ М/с) ^{P ( X i ,}₁ - F j( c ) ^• ^{X k )} ’ where Fj(c) is given by (1.6).

The function p *(x ly xk) satisfies the latter of conditions (1.2) since Fjic) < 1 {c < n).

In the Polya distribution (0.1) we restrict now the values of t random variables, e.g. X ly X 2, X t (t < h) from the left-hand side by ciy

i ~ 1, 2, t, i.e.:

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where cf must satisfy the condition

(1.Ю) + c2 + • • • + ct < n . With regard to the condition

^ 1 + ^ 2 + ••• = П

the left restriction for the values of the random variables _ЗГг- by c{ (i — 1 ,2 , involves the right restriction for other random variables

For the given conditions, in the analogous way as formerly, we deduce the formula for the probability function of the truncated distri

bution of the joint random variable (X x, X 2, ..., X k).

Denote by X k+ln the class of all systems (xx, x 2, . . . , x k+1) of fc-f-l non-negative integers satisfying the conditions

0 ^ xi ^ ^ti — (cx 4~ c2 + •.. + ct i — t + 1, t + 2, ..., It + 1 ;

#1 + #2 4“ • • • 4“ #A+1 = M

and by K k+l n the class of all systems of Tc+1 non-negative integers satis

fying the conditions

0 ^ xi — (Cj -J- c2 4~ • • • 4- , i = 14-1, t -{-2, ..., Jc 4-1 •

c{ + 1 < x{ < n , i — 1, 2, ..., t ;

0 < xt < c , i = l , 2 , . . . , < ;

0 4 = / + 1 , /-j-2, . .., f t + l ; 4-^24- ••• 4-^a+i — w.

With these notations we can write

With respect to condition (1.10) and relation (0.4) we have

^ k + 1 — t,n—t,n—(xx + ... + Xi) (jB14-... + arp

-j^[n, — a]

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Multidimensional truncated P olya distribution 195

Let us note that for t < к and 0 < x{ < n, i = 1 , 2 , . . . , t

(1.11) .... «(«i)

nlxi+-+x0 it».-a]

t

П_{i«= 1} ^V^{[£сг-.-а]}ⁱ

(- i »

[n -(x x+ .. .+ x t) ,- a ]

is the probability function of the t-dimensional marginal distribution in the Tc-dimensional Polya joint distribution.

Analogically as formerly

ci... ct

(1*12) F 1% ^t(cx, ..., ct) = ^ Pi,...,t(æii • • • , ®t)

x x,...,Xf=>0

is the value of the distribution function F Xi^ t(xx, ..., xt) of (1.11) for Щ = Gi> i = 1 , 2 ,

Taking into account relation (1.12) we obtain

£ P(%i>-” >®л) = l ~ F i , . . . , t{cx, . ..,c t)

^fc+i.n hence we obtain

Th e o r e m 1.2. I f we are given probability function p(xx, ..., xk) of the h-dimensional joint] random variable (X x, ..., X k) of the Polya distri

bution (0.1) and the values oft variables, e.g. X x, ..., X t, are truncated below by cx, . . . , c t, then the probability function p*(xx, ..., xk) of the obtained truncated distribution is determined by

(1.13) p*(æx, . . . , x k) p(xx, . . . ,xk) 1 ~-^\,...,t(Gl, Gt) , where F x^ t(cx, . . . , c t) is given by (1.12).

Since F Xi tt(cx, ..., ct) < 1 (because of ci < n for i = 1, 2, ..., t) then p *(x x, ®ft) > 0 and thereby satisfies the latter of conditions (1.2).

As the special cases of the truncated Polya distribution we have:

the ^-dimensional truncated Bernoulli (binomial) distribution (if a = 0) and the k-dimensional truncated hypergeometric distribution (if a = —1{N, Pi = NpilN).

According to (1.8) and (1.13) the first distribution has the forms respectively :

p * ( æ x, ...,œk) 1

----:--- П l - F j ( c )

П^i=s:k+1¹ ^Pi_^xA ^,

(1.14)

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where

Щ °) = £ ( ^ ? ( 1 - з > / Л

&+l (1.15) p*(œx, . . . , x k

where

F Gt) -

' Ш ’

1 _{~ F x}₉ ₉_t(G11 •_{••1 Gt)} _г—1

2

....

ⁿ

ж1,...,а ^ = 0 г = 1 ‘ г=1

n - (x x+ ...+ x t)

The probability function of the dimensional truncated hypergeometric distribution takes the forms respectively:

/Тvr dxi\

(NPi) г

(1.16) ^{- , x k)}⁼ 1 nl

l - W ^{N [n]}^J₁ where

Щ ( с )

C

= V

Xj = 0

/ « \ (JV>3-)[iBd

\ficy/ jyW ^{' W} ^-

(1.17) p *(x x, . . . , x k) 1

~ 1 - •J Gt)

where

• • ч ct)

П ^SDA

nl ^k+1

П^(ffipi)^[Щ]

-

¹ ^‘ П^A^t^{№ ,.)}^x, ^m ^t [n-(*i +—+ap]

From (1.8) and (1.13) we obtain the probability function of the truncated distribution connected with the generalized scheme of Polya if a = s/N, Pi = N p JN :

(1.18) p*(Xi, xk) =

• k~{~ 1 ч —s3 nl T~f { ^ P i ) 1 1 -F A c ) N [n’~s] i 1 Пi —X a?,!

where

Щ с) =

, [ X j , - S ]

^ (n\ (ÏÏPj) Г ^{, лт} -хг„ \ln -X j,-8 ]

(N — ЖрЛ

(1.19) Р * { Х ц • • • j ^k) —

Xj) j y [ » . - s ]

1

1 - F x_ >t(cx, . . . , c t) n\

I Iг=1 flb!

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where

? • • •} ct) ci>‘-'ct

I П[^i+ ...+*<] *

]f[n ,-s] П1 = 1

№ < )^{[xit- s} ^t X,-l N -■ N Yjs г=1

[n -(X ! + ... + « / ) ,- «

Knowing the probability function of the fc-dimensional joint random variable (Х г, . . . , Х к) it is possible to determine the conditional distri

bution of the random variable (X t+l, ..., X k) assuming that other t ran

dom variables have taken the values (xt , . . . , x t).

The conditional probability function is given by

(1^.20⁾ P (щ+l i • • ч æk I » • • • > ^t) Р(яи •••>%)

We determine now the probability function of the conditional Polya distribution.

After using formulae (1.20), (0.1) and (1.11) we obtain

Р(Щ+\1 • • • ? I ^ 1 ) • * • 7 ®t)

f ** * +1 J -Xi>~a\ P 1 it/, Л х1+-'-+Ч\ ”* Г 1 „ [ *» > - « ]* ]{n,-d\

X

Пi—l

( i - j »

X, ^{r^[n,-a]} n ^ . x

[n—(Xi+...+Xf), —a]

i= 1 nl

[Xj + . . . +Х^]

П

&+1 [xj,—a] , t

n v / i ' - a

\[w - ( x i + ...+ x ^ ) ,- a ]

t=l

i=t + 1 With regard to the condition

xt+l-\-xt+2-\- ... -\-xk + l= n — (x1-\-xi -{- ... -\-xt) we obtain finally

Th e o r e m 1.3. The conditional probability .function p (xt+l, ..., xk | xx, ...

> ..,x t) of the joint random variable (X t+l, ..., X k) in the k-dimensional Polya distribution (0.1) of the random variable (X l f . . . , X k) assuming that other t random variables X x, ..., X t have taken the values xx, . . . , x t, is given by

fc+1 [tff.-a]

(<*7+1+ ••• + ®*+l)’* f ] ~ Y f -- i=t+l

(1.21 ) p {x ^ i, • •., Xk I Xx, ..., Xt) —

(Pt+i + ••• P P k + i)И+1 + ...+жл+1,-а1

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Using formulae (1.20), (0.1) and (1.5) we determine the conditional probability function p (x x, ..., Xj_x, xj+15 xk \ Xj) following as before:

P(x i, ^, ^{1 ,}^{X j}+ 1 , • • • ) X k^IX j ) р {хг, . . . , x k) n

pi”

PjiX j) fc+I [x^,—a]

* ! r j P i ' / М P i ’ ’

~a] 1 1 »__i xA /* W

lX j,-a ]

a - V j ) 2 [».-«]

[n —X j,— a]

= (n xi)' j [

&+1 [Xi,—a]

Pi i=li¥=j

X^. ⁽¹~ P j)^{[n —}^Xj,—^a]

Thus we have

Th e o r e m 1.4. The conditional probability function p {x x, ..., Xj_x, xj+ l, . . . , x k \ Xj) of the joint random variable (Xx, ..., X j_x, X j+1, ..., X k) in the k-dimensionad Polya distribution (0.1) of the random variable {X x, ..., X k) assuming that the random variable X j (1 k) has taken the value Xj, is given by

fc+1 Ащ>-а]

П ——

1 1 X{ 1

(1.22) p {x x, ..., a>j_x, xj+1, ..., xk I Xj) = (n - Xj\\

i=iгф]

' (1 - P j) ln~Xj’ ^

The analogous theorems can be written as corollaries for the bino

mial and hypergeometric distributions.

2. The factorial moments of the multidimensional truncated Polya distribution. Let the classes K k+1>n, K k+l n, K k+l n have the same meaning as they had at the derivation of probability function of the truncated Polya distribution (1.8).

Th e o r e m 2.1. The factorial moment a*ri.... of order r — rx + ... + г к

of the k-dimensional truncated Polya distribution (1.8) is expressed by

(2.1) where

Vl: ■г*] ~ alrv E r*] 1 - F j{ c ) ’

E 1 -

(1 + ra)[n~r’- a] ² ^{( r}

1 = 0

(PjJr rj a)^’ 0] x

x ( l — pj + a(r — г5))[п r *• a]

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and

a lr1; <rk\

n[r] l~Jpli ù a]

i= 1______

2 [r,-a\

is the factorial moment of r-th order of the Polya distribution (0.1).

P roof. According to the definition of the factorial moment (0.2) we have:

a^*_{h i,} A W ■. . . X lk k]p * { X !, . . . , x k)

Kk+l,n

1

1 ² г " . . . æ[k k]p ( œi, •• • ? x k.

K*+l,n

Making use of property (1.1) we can write

к

1/1....rfc] 1 -F A c ) \

3 Kk+l,n ^

1 1

;+ l,n t==1

к

•••,* * ) ) = 1 3 г Г (3 )'( Л _ 'В)'

ck+\,n i2 = 1

The value of the term A has been calculated in [1] (p. 47, (3.1)).

It was the value of the factorial moment of order r = rx -f r2 -f ... + rk of the Polya distribution (0.1):

(2.2) ^ — ahx. •••.>•*]

mw f ] p p - a' i=*l_____

p r ,- a ]

We will calculate the value of the term В with analogous argumen

tation as in the previous section. We have namely

*+i

* - ¹ /⁷ ^{X {}~M^{p [«»-«]}^{]~ J[}^P'^[хг-,-й]

K-k+l,n ^{г = 1}

y i X^L ^_ [ X j , - a ]

/ , xAI Pi

X i = 0 J

* Jrü r . J Xk+1’~a]

_____ ^{I /} ^{' Ч .}_^A X i ’ ~ a ] jPfc+1 ltn.-а] j J j Pi

ir E i= 1 ^X^fc+1;^Î

where the class K k n_x. has the same meaning as at the deducing of for

mula (1.8).

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f r .1

It follows from the definition of the factorial polynomial'that x 3 = 0 for Tj > c and thereby В — 0. Then assume that г,- < c. With respect to — 0 for ooj < Tj we shall sum up over xj = r^, ..., c.

Let K'kn_x. c= K kn_x. be the class of all systems {хг, ..., Xj_Y, xj+1, ...

..., ock+l) of Ji non-negative integers such that

••• + ®j+i “h ••• “b^A+i = w — Xj, r ^ x ^ n - X j , i = 1, ..., j - 1 , j + 1 , fc,

0 < xk+1 < ю —a?,-.

The summation over the class K k>n_x. in В can be replaced by the summation over the class K kn_Xj since x[p ] = 0 for xi < r / , i = 1, ...

..., j — 1, j + 1 , ...,1c. If, moreover, we take into consideration the follow

ing equalities resulting from the definition of factorial polynomial Р? {- а1 = p ? p-* { p t + rt* f ™ - * ,

(2.3) X[*•<]

х{1 ( ъ - г {)! ’

Xln.-a] ==1 [r,-a](1 _|_ra)[»-r,-a]j then we can write further

* , . [ r * - o ]

, — ( /n^Lor . n.\ lX3~rj>

W^[r] П р<"

в 1

(n - r У 3 р (Р, + ^ а )1

x 2

^к,п—хл

u'3~’3

(n — r — Xj + Tj) ! p'jk+t1’

r,.)!(l + m)[n- r’- a]

I -,[**+1>-«] k x

'fe+1 Пi= 1

гФЗ

(Pi + r{a) {щ -и У

Put xi — ri = т {, i = 1, ..., j — 1, j + 1 , ...,1c-, xk+1 = mk+1. The con

dition

Хг+ . . . + ®j-i + Xj+\ + • • • + Xk+1 — n — Xj is taking now the form

% -f ... + » Н + Ют + ... + mft+1

= - (» - !+ ... + r j_ x + rj+1+ ... + r k).

The above relationship can be written in the form (2.4) m1-f- ... + » » н Л-Щ+i + ... + mA+1 = n — xj — r + rj since

rj-f ... + r M + r y+1+ ... + r k = r - r , . .

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The class K k>n_x. goes into the class K k>n_Xj_r+r. of all systems (m1? ...

wij^, mj+1, mk+1) of Jc non-negative integers satisfying condition (2.4).

Using the above notation we can write

В = j j / ï * ' * 1 ( п - г р - ’>\Ь + г ,а р - Ч ’- а)

Ur>~°] (1 + m)[n r’ — ! x

x у ( n - V j - r + r^l pl£*j+1, a] А {Рг + г{а)1т{г■

Z^j mk+l\ 1 1

- a ]

г'=1

i ФЗ

т л

K k,n—Xj—r + r j

Making use of the Newton formula (0.4) we obtain

» w ”

г—1___________ \ ’ _v_"'_____________________________________

l [r Zj (1 + ra)^n~r,~a^ (Xj — гЛ !

• nr..-—Г’ J J

(■n — r)[Xj ^{P j + Tja)^ rj’ a]

X

x(Pi + ••• + ib -i+ Ib + i + ••• +^fc+i + a (ri + ••• + U -i + ri+i + + ... + rk)fn- r-x^ ’- a 1 A (n — r p - r* (pj + Tja)[Xj~rj’~a]

= alri,:;rk] j ( г + г а ^ - ^ Ц х t- г,•)! X

xn ri 1

x( l- р , + а (г - г ,))[п- г- х^ - а\

In account that Xj takes only the integer values we put Xj — rj = i, i = 0,1, ..., c — rj and obtain finally

C-r„-

^ ( n - r ) [i](pj + rja)li‘ a]

(2.5) В = a[ri,...,rfc] > . ... .... , --- X Z^j (1 + ra)[n~r,_a4 !

x (1 — pj + a(r — Гу))[п_г_г’’_a].

Considering equalities (2.2) and (2.5) we have

1 vi* (n — г)и (р,- + г7-а)[г,-а1

ai — * 11--- x -

Vl.... 'ki X - F ,{c) l1 (l + ra)‘n r ,- a ] ^ / j^{___ y} X l\

X (1 - p , + a(r Я

and thereby the theorem has been proved.

Next we deduce the formula for the factorial moment of r-th order of the truncated Polya distribution (1.13). We assume for this purpose that the classes K k+l n, К к+1гП, K k+l n have the same meaning as they had at deducing of the formula of the probability function (1.13).

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Theorem 2.2. The factorial moment ° f order r = rx + ... + rk of the k-dimensional truncated Polya distribution (1.13) is given by

< 2 -6 ) * .... * ^- . - * •^... ^X

X 1 -

Aix+...+iu

1 s r \ { n — r)

(1 + ra)ln~r'~a] Z^j i±l ... itl

X f j (Pi + r j a p ’ ”‘( l - ^ P t + a ( r - ^ r/j)

3 = 1 ' 3 = 1 3 = 1

assuming that c r { for i = 1, 2, ...,t , where

[n -r-{ii+ ...+ it),-a]

»wг=Х p r . - a ]

is the factorial moment of r-th order of the Polya distribution (0.1).

P roof. The theorem will be proved in analogous way as in the demonstration of Theorem 2.1. We have namely

v ^...^rk]= ^ 4 ri1 ...a% dp*(x1, . . . , æ k)

J T œl[ ù . . . x [^k]p(œlf ...,<ck) Â:+l,n

1 F 1...t(C 1? * * * ^?Ч) к к+1,п

К

--- ^ ---( У1 ] ~ 1 (^î?1 - F 1 t(Cl, . . . , c t) \ Z j 1 1 } L’ ’ к’

к

- J ? f ] 4 j]P (æl ’

_" А_1 '

К 'к+1,п j= 1

hence (2.7)

[ri r‘ 1 1 --P , ,(Cl

The value of the member A is as before equal to лм П р [р ’~~а]

(A - B ).

(2.8) A = -pr,-a\

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We calculate now the value of the member В :

В = [ J («!> •••> Xk)

K'k+l,n 3 = 1 ^{■— У}

cl ' - ’ct t [r^A [ x j f - a ]

Xj 3 Pj 3 1

~œA l [w’~a]

2 П

x l t . . . , x t = 0 j= 1

X

X 2 ^ П

Jb'1 [rj.-a] [xk+1 ,-a]

P j Pk+l

'-fc+1—t, n — ( x i + . . . + x t ) j= t+1 аь! J a?fc+i ■

where K k+1_tjn_{Xi+i^+Xt) is the class of all systems {xt+1,x t+2, . . . , x k+1) of k + l — t non-negative integers such that

X t + l + X t+2 + • • • + # f c + l — n — ( x 1 Jr . . . + X t ) .

rr .i

It follows from the properties of the factorial polynomial that x)3 = 0 for Xj, C f j, j — 1, 2, ..., Tc. Thus in the first sum xj = rj7 r,- + 1, . cj7 j = 1, 2, ..., t and in the second one the class K k+1_tn_^Xi+^+Xt) can be replaced by the class K k+l_tn, denoting all systems (mt+1, ...,m k+l) of k + l — t non-negative integers, where xi — ri = mj7 j = t + l,...,T c and mk+1 = xk+1, such that

Щ+1 + Щ+2 +^{. . .} + mk+1 = n', where

(2.9) n' = n — {x1+ . . . + xt) - ( r t+1+ ... + r k).

Taking into account the above properties and also relations (2.3) we can write

В = []p]j>

3=1_________

ph»-o]

x x= r b ...,x t ^ r t

(w_ r)[»-»'-rl (1 + ra)[n~r>-a] x

x П^{(Pj + rj}^a)[ X j - r j , - a ]

(Xj rAl Kk+l-t,n ’ 2

Л

3=t+1

(Р з+ Гза)[ m j . - a ] [m k + v - a

Pk+l

m mfc+i'

Making use of the generalized multinomial Newton formula (0.4) and also of

(2.Ю) r<+1 + r <+2 + ••• + r k = r — (r! + r2+ ... + r*)

3 — Pra c e M a te m a tyczne 19 z. 2

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we obtain

B alrv-,rk] (i + m)1”" ’'*- ®1

3!! = ^ , . . . , ^ = ^

On the ground of relation (2.10) equality (2.9) is transformed in (2.11) n' = n — (жх+ ... +£»*)—r + ( r i + ... + r*).

The variables Xj ( j = 1 , 2 ,...,< ) take only the integer values, then with respect to (2.11) we can write the following equalities:

n — n' — r — (a?x+ ••• +»<) — ( ^ + ••• + »*) = Ь + г2+ ••• + ^ , u/ = w — r — (ijd- ... + q ) .

Making use of above equalities we can write

After substituting in formula (2.7) instead of A formula (2.8) and instead of В formula (2.12) we obtain the required formula (2.6).

Corollary 2.1. The factorial moment a*ri,...,rA] °f order r = гг + ...

of the h-dimensional truncated binomial distribution (1.14) and (1.15), respectively, is given by

after substituting ^oD^j — V^j = ij, ii = 0,1, ..., C^j — V^j, j = 1, 2, ..., t, and

c1- r l,...,ct- rt

( 2 . 1 2 ) В a [r 1, . . . , r fc]

I ( l + r a ) [n~r>-a]

гъ ...,ц = 0

[ n - r - ( t 1+ . . . + f p , - a ]

(2.14) a * 1( a [ r i , . . . , r A]

n - r - O i + .-.+ t f)'

к where

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P roof. It follows immediately from (2.1) and (2.6) if a = 0.

Co r o lla r y 2.2. The factorial moment a*riJ...>rA] of order r = r1-{- ...

... -\-rk of the h-dimensional truncated hypergeometric distribution (1.16) and (1.17), respectively, is given by

,01K, » “ l'i....*•*!

<2-15) а 1ч....r * ] = ~ ----= — II

where

l — Fjie) \ (Ж -г) {n—r] ^X

X S n ^ )U (N pj- га.)[г'](Ж- Nps- { г - гу))[п” г_*] j ,

г —0

<2-16) 4 , ... . =

a[»•!....rk\

[ri....r*’ 1 -- P ,... ( C l , - , 4> ^X

cx- r x,...,ct-rt

x | 1 ~ ( j _ , ) [ ■ . - ) У , ( « - > - ) |i,+- +i,1x

m •••»*< =о (N — r

3 = 1 3 ' j= 1 " - 1 '

» [1 П (Я р {)™

t i=1

[ п - г - ( г 1 + ...Ч -ед

З т [г]

P roof. It is evident immediately from (2.1) and (2.6) if a = —1/N and ^ = Np{jN.

Co r o l l a r y2.3. The factorial moment a *r i ,...,r k] ° f order r = гг+ ... -\-rk of the h-dimensional truncated distribution (1.18) and (1.19), respectively, connected with the generalized Polya scheme is given by

(2.17)

(2.18)

[ri.... r*] l - F j ( c ) \ ( N p s r f n- r>-s~l c_rj

1

X / Wi _ [^*]

x > ---- ---(Npj + svj)[l,~s](iV — Npj + s (r — ri

г=0

a [rv ...,r k\

1 - F 1 J c 1, . . . , c t) X

C i- r lt...,ct- r t

X 1 (Ж P sr)[ n —r, —s] 2 1

*1 ,...,i j = 0

[n- rf l+- * iù X

t - t y . - e ]

П3 = 1

{Npj + srf) r U

* I \ \ [ n - r - ( i 1 + . . . + i t) , - s l \

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where

«[га, rfcl

» w n w p f * " ' i=l_________

J ÿ l r . s ]

P roof. We obtain the above formulae from (2.1) and (2.6) if a = s/N and Pi = N p JN .

3. The ordinary moments of the multidimensional truncated Polya distribution. Let 8$ be Stirling numbers of the second kind, i.e. the coeffi-

cients in the identity: xr = ^ 8rtx[l\Г 1=0

Theorem 3.1. The ordinary moment of order r — ... -\-rk of the h-dimensional truncated Polya distribution (1.8) is given by

m < ... - * “ 1^ I . П 8>

3 lv ...,lk=0 1=1 x ah.... h 1 -

k t»“ 2 Ц,~а\

(i + « S h ) i = l

X

c—h_{J? ( n} [П - 2 l j - i , - a ]

X \ --- ---(Pj + l j a ^ ’ - ^ l - p j + a i r - l j ) ) 3=1

where

[ .2 li] k U- a\

n v f - 1

_ i = l

ah,...,lk - к

[ 2 к , - a ]

i=l

P roof. We use the formula ([1], (4.2), p. 49) expressing the rela

tionship between the ordinary and factorial moments (3-2) W...,rk = ’j f f ] s : i a [h....Щ,

ll,...,lk=0 1 = 1

where ar >r and is the ordinary moment of order r = гг -\- ...

... ~гтк and the factorial moment of order l = l1-{- ... + l k respectively.

For distribution (1.8) formula (3.2) takes the form

rl к

i .-.»•* = £ n ^ li a*h.... lkP h>-.-ik-0 i=1

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Calculating а*г^ ^ from (2.1) we obtain formula (3.1).

Th e o r e m 3.2. The ordinary moment a* ,r of order r = гг-{- ... + r k of the к-dimemional Polya distribution (1.13) is given by

(3.3) a * . =

(c 1, . . . , ^ 1 X

® h a lh>-d k l I1

ri >->rk к

X h’---dk = 01 П t=1

Cl- l v ...,ct- lt k t dh,-a]

x 2 [п ~ Ё Я ' / 7 -(ft w

к к t«- 2 h.-a]

(i + * Z h ) •“

i=l

X

t

i=l i=i b'! ^X

(> к k

( V i vn Uw_ 2 h~(4+...+ч),-°1\

- 1

i=l i=l

P roof. We proceed analogically as in the proof of the former theorem with this difference that we calculate а*г г j from (2.6).

In the case of the jc-dimensional joint distribution the ordinary mo

ments of the first order are equal to the expected values of the corre

sponding marginal distributions of random variables X { (i — 1, 2, ..., Jc).

We obtain the formulae for the expected values of the truncated Polya distribution (1.8) and (1.13) from (3.1) and (3.3) or directly from (2.1) and (2.6). (The ordinal у and factorial moments of the first order are equal.)

In the case of distribution (1.8) the expected value of the random variable for which the truncation has not been made, i.e. for X where i = 1, 2, ..., j - 1 , j+ 1 , ..., fc, is given by

(3.4) E {X i nPi

1 -F A c ) 1 - Ii = 0

(n -1 ) [<]

P[b-o]^X

"while for the expected value of the random variable with truncation we have

(3.5) F iX j) = nPi 1 - Щ с )

1 у 1' (n— 1)^

_____ 2^ Tj—

х(^- + а)[г'’ а1(1— Pj){n 1 *’ al

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In the case of the Pôlya distribution (1.13) if we investigate the expected value of the one of the random variables for which the trunca

tion has not been made, i.e. for X {, where i = t +1, ..., k, then we obtain

(3.6) E (X t) = nPi 1 -

cV - ’ct m 1 + ...+it] t t

sp {n —ly 1 —

X

Ù ! . . . г, !

»'l.... *<=0 j = 1

Pi> ( i + « - 2 f t )

,=i '

while for the random variable with truncation, i.e. for X f , where i = 1, ..., t, is

(3.7) Ï ( I , ) = - --- nV‘ ^{п р {} 1^! 1

. , t ( c и • • • :, « , ) \

X У

Û...г'г>---Л =°

x / 7 p / V ’ - a ] ,

X

(n — l)Pl + -+^]

t i=ij¥=i

(Pi + U )1**’ a]^X

)■

\[n -l-(i1+...+ie),-a]

i=i

The analogous formulae can be written also for the binomial and hypergeometric distributions. Namely —

b in o m ia l:

(a) without truncation for X { (i = 1, ... , j — 1, j + ! , . . . , & ) (3.8)

1 - Щ с )

а д = v - ^ ( x -

ÿ

¹ - л ) - — ),

i=0 %\

(b) with truncation for X j

(a') without truncation for X { (i = t-fl, , ft) (3.10) Щ Х {) пр{

1 • • • j

X 1 X

2 ;'

(W— 1)[г‘1 + --- + г‘Л

П^{^} ^x

j'=i

x

L VI \»-i-(<!+...+f,)\t

i=i '