The moments of Pólya distribution. Special case

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I I (1969)

Wacław Dyczka (Łódź)

The moments of Pólya distribution. Special case

Introduction and summary. The paper contains the evaluation of the ordinary factorial moments (5) as well as that of the ordinary moments (13) and the recurrence formula for the ordinary moments (21) of Pólya distribution (4). This distribution is more general than the distribution of Pólya (6) considered in [3], and it has been called in this paper the distribution connected with the Pólya urn scheme. Its greater generality consists in p being an arbitrary real number from the interval (0 ;1 ).

As particular cases of the formulae obtained I got the corresponding formulae (a majority of them are known) in the binomial distribution (p is arbitrary not necessarily rational), in the negative binomial distri­

bution, in Poisson distribution, in the hypergeometric distribution and in the distribution connected with the urn scheme of Pólya. It seems that the obtained formulae are simple. In the paper use is made of the properties of factorial polynomials and of the recurrence formula (22) for Stirling numbers of the second kind. These numbers may be evaluated by means of this formula or read from tables which may be found in text-books of numerical methods. The notion of finite differences has also been used in the paper.

§ 1. Ordinary factorial moments of a random variable with Pólya distribution. Particular cases.

Definition

1. A polynomial of the form

in which h is an arbitrary real number is called a factorial polynomial with respect to x of the degree r (or the generalized r-th power of x).

If h — 0, then = xr which justifies the use of that symbol. If h = 1 it will be written simpler = x[r\ Moreover, we accept that = 1 for x Ф 0. For the generalized power with any In the formula

(1)

x' = x (x — h)(x— 2h)‘...-[x — (r—l)h ],

Roczniki PTM — Prace Matematyczne XIII 9

130 W . D y c z k a

is true [5]. Formula (2) will be called the generalized Newton binomial formula (it becomes the ordinary Newton binomial formula for h — 0).

D

2. The expression oo

(3) a[rj = J x[r]dF(x)

is called the ordinary factorial moment of order r of the random variable with the distribution function F(x).

D

3. A random variable X has Pólya distribution [2]

and [7] if its distribution function is given by the formula

(4) P (X = m) = F ...--- ( w - 0 , 1 , 2 , . . . , n),

where p , q, a are arbitrary real numbers (not necessarily rational) satisfying the conditions: p > 0, # > 0, p-\-q = 1, n (—a) < m in^, q).

Theorem 1.

The ordinary factorial moment o f order r of a random variable with Pólya distribution is given by the following formula

(5) a[r] = (r = 0, 1, 2, ...).

P ro o f. By Definition 2, the formula for the ordinary factorial moments of a random variable with the Pólya distribution is

(a) pw.- a]

If r > 7i, a[r] = 0, for w[r] = 0 if m = 0, 1 ,2 , n and r > n.

Thus in this case formula (5) is true. Let further r < w. Since m[r] — 0 for m = 0 , 1 , 2 , . . . , r —1, (a) takes the form

(b) 1

а[Г] — yfw.-a]

Taking into account Definition 1 we have

( c )

= p lr’- a](p + ra)[m~r’- a\

( d )

mИ ( 1 ^\mj = ^{mm wm — m[r] mlr](m—r)!}

( e )

ml

Substituting (c), (d), (e) into (b) we obtain

( f )

a[r] — ^ V r' " a]

l [r- a](l + ra)

2 ( m - r ) (

Н-™)1^ - V 1- ' - 4 . I t remains to be proved that the sum in the last formula is equal to (l + ra)[n~r’-al. After substituting in this sum m — r = l, in view of

|or we obtain by (2)

у I n —r X

\m—r

m= о \

(р + га)1т- г-~“ 2

(V)

{p + ra + q)[n' r - a] = (l + ra)[n- r*-a].

Formula (5) is not new, it may be found in paper [4]; there it has been deduced by another method, applying hypergeometric series. a[rJ is then the value of the sum of that series (which in case of the Pólya distribution is a finite sum). In this paper the formula for a[r] has been obtained in an elementary way, using only the generalized Newton binomial formula.

D

4. X is said to be a random variable connected with the urn scheme o f Polya if its distribution function is given by the for­

mula

(

6

P (X = m) \m] jy[n, - s]

where s is an arbitrary fixed integer, N — the number of members of the universe, M — the number of members of the universe which possess the specific characteristic, n — the number of such successive random choices, that after every random choice of one member of the population that member is returned and according whether s ^ 0 or s < 0, s elements with the characteristic of the chosen element, are added to or subtracted from the population; m — the number of elements in the sample with the size n possessing the specific characteristic. If s < 0, we assume that

— sn < min(M , N —M).

C

1.1. The ordinary factorial moment of order r of a random variable connected with the urn scheme of Polya is given by the formula

(7) a[r] = n[r]M^~sf N [r’~s].

P ro o f. A random variable with Pólya distribution given by for­

mula (4) becomes a random variable connected with the urn scheme

of Pólya if in formula (4) we put p = M/N and a — s/N. Substituting

these values of the parameters p and a into (6) we obtain (7).

132 W. D y c z k a

C

1.2. The ordinary factorial moment of order r of a random variable with the (ordinary) hyper geometric distribution [3] is given by the formula

(8) a[r] = n[r]M[r]IN[r].

P ro o f. The random variable with distribution (6) for s = —1 is a variable with the hypergeometric distribution, therefore putting s = — 1 in (7) we obtain (8).

C

1.3. The ordinary factorial moment of order r of a random variable with binomial distribution [3] is given by the formula

(9) a[r] = n[r]p r.

P ro o f. The distribution function (4) tends to the distribution func­

tion of the random variable of Bernoulli as a -> 0, therefore passing to the limit in (5) as a 0, we obtain (9).

C

1.4. The ordinary factorial moment of a random variable with Poisson distribution [3] is given by the formula

(10) a[rj = /f, where Я = np.

P ro o f. L et in formula (9) n -> oo and lim np — Я > 0, then n-> 00

n[r]p r — Пр ( Пр — рЦ Пр — 2р)-...-[п р — (г—1)р'] -> {n p f — Яг.

De f i n i t io n

5. A random variable X has a negative binomial dis­

tribution [3] if its distribution function is expressed by the formula (11) P ( X = m) = ( - l ) “ ( ~ ”) J> V (m = 0, I, 2, ...),

where p , q ,v are fixed numbers and satisfy conditions p > 0, q > 0, p + q = 1, v > 0.

Co r o l l a r y

1.5. The ordinary functorial moment of order r o f random variable with a negative binomial distribution is given by the following formula

(12) = { v p l q f - m = (p/s)V ' - 4 .

P ro o f. I t is know [2], that a distribution function given by (4) for (g) n o o , p -> 0, a -> 0, np -> Я > 0, п а -> 1 /д

converges to distribution function (11) with the following values of the parameters

(h)

2 = e/(i + e).

The limit of (6) under conditions (g) is obtained as follows:

a[r] = n[r]p [r’~a] l l [r’~a] = {np(np-\-na—p — a )’...-\_np-\-(r—l)na-\-(r—l) p —

— (r—l)2a]/llr'- al} -^Я(А+1/е)(А + 2/г)-...-[А + ( г - 1 ) / е] = A''--1"'.

Taking into account that

= qjp and A = pv/q we obtain formula (12).

Observe, moreover, that formula (10) may be obtained from formula (12) for ([3]) if

(i)

and vpfq = const > 0,

then distribution function (11) converges to the distribution function of the random variable of Poisson with the mean value vp/q. Thus to obtain formula (10) it suffices to take into account conditions (i) in (12).

R em ark 1. Formulae (9) and (12) may be found in [5]. The fac­

torial moments have been evaluated there by the method of the gener­

ating function. Formulae (8) and (10) can also be found in [1] (in the problems to be solved). The above formulae (8), (9), (10) and (12) have been obtained in this paper as particular (or limit) cases of the more general formula (5).

R em ark 2. For the ordinary factorial moments in Pólya distribu­

tion the following recurrence formula

(5') a[r] = {{n — r + l ) [ p + {r — l)a]/[l + (r—l)a ]}a [r_ 4

is true. This follows immediately from (5) and from the definition of the factorial polynomial.

The following recurrence formulae for the ordinary factorial mo­

ments in the distributions named in the Corollaries 1.1-1.5 follow from (5'):

a. for the distribution connected with the urn scheme of Pólya (7') аи = { ( n - r ± l ) [ M + ( r - l ) s ] l [ N + ( r - l ) s ] } a lr_lV,

b. for the hypergeometric distribution

(8') О.Щ = {{n — —r -p 1)} «[r _ i] ?

c. for the binomial distribution

(9') a[r] = {n— y + l)p a [r_ 4 ; d. for the distribution of Poisson

(10') а[Г] = Яа[Г_1 j*,

e. for the negative binomial distribution

(12') a[r] = {plq)(v + r —1)а[г_ ц.

134 W . D y o z k a

T

2. The ordinary moment of order r of a random variable with the Polya distribution is given by the formula

§ 2 . The ordinary moments of Pólya distribution. Particular cases.

(13) 1

1 = 0

S ’ w[ЧрР,-а]

lP .-e]

where SI are Stirling numbers of the second hind.

P ro o f. Stirling numbers of the second kind are defined as the coefficients of a factorial polynomial in the identity (there are also other definitions of these numbers):

(14) xr = ^ + ^ я ? [1]+ ^ ж [2]+ ... + # я ет.

From (14) it follows that Si — 0 for i > 0 or i < 0; S* = 1 for r = 1, 2, . .. ; = 0 for г Ф 0.

In view of (14) and of the definitions of moments ar and a[r] we obtain a general formula (true for an arbitrary distribution in which the corresponding moments exist):

O O 0 0

r r

r

ar = f xrdF(x) = f £ s riXli]dF(x) = J T # J x[i]dF{x) = J ^ S la ^

— oo —oo

l=0 i=0

thus Г

(15) ar = y « a (<].

i=0

After substituting (5) into (15) we obtain (13).

C

. The ordinary moments in the distributions named in the Corollaries 1.1-1.5 are given by the form ulae:

a. fo r a distribution connected with the urn scheme o f Polya

(16) ar

Г г=0

n[i4 l [i’~s]

»

b. for a hypergeometric distribution

(17) Ctr — ^ j

г—0

N \i) ’ c. for a binomial distribution

Г

(18)

i

1=0

d. for the distribution of Poisson

r

(1 9 ) a r = ^ $ i A ‘ ;

1 = 0

e. for a negative binomial distribution

(2 0⁾

Formulae (17), (18) and (19) have been obtained by another method in paper [6]. I have obtained them above as particular cases of the new formula (13), which in its turn has been obtained by means of factorial moments.

§ 3. The recurrence formulae for the ordinary moments of Pólya distribution. Particular cases.

Th e o r e m 3 .

For the ordinary moments of a random variable with Pólya distribution the following recurrence formula is true

(21) ar+1 = npar{n — l , p + a, 1 Ą-a) + {p{a) Aav ar,

where ar = ar(n, p , 1), A^ar is the first finite difference in relation to p o f the moment o f order r with the step а Ф 0.

P roof. According to formula (13) we have

For Stirling numbers of the second kind the following recurrence formula is true

(a)

(

22

$ i+1 — {f — 0 , 1 , 2 , . . . ) . Substituting (22) into (a) we obtain

(b)

136 W . D y c z k a

for = 0 for i < 0. Putting i —1 = l, we have further ( w - l) tZ](2> + a)[*’_a]

(с) В = n p £ Si

?=o (l + « npar(n — 1, p-\- a, 1 + a).

Now we will find the first finite difference in relation to p of the moment

with the step

0

Zl“ a, =

2 i=0 : ^{Si wW}

' /Iеn vP >

since

A l p [i~a] = (p + a f> -a]- p [i’~a] = (a/p)ipli’- a]

(taking, moreover, into account that $£+1 = 0)

a . r n W ^ X a r = — % p[t-Q]

whence

( d )

A = (p / a ) Apdr.

Taking into account (c), (d), in (b) we obtain the required formula (21).

Writing formula (21) in the following equivalent form Or+i = [(npI%)ar(n—l,p -\ -a , х-\~а)-{-(р1а)Араг(п, p, x)\x=1

we will find the first three ordinary moments in the distribution of Pólya Щ>_ . i + J L ( i _ i ) l _ nP

x a \x==1

a, =

= i ; « . = [

np (n —l){p-\-a) np{p-\- a —p)

x = n p ;

xA-a +

ax ]a:=l

« 3 =

n (n —l)p{p-{- a) np

x(x-{-a) ж

n (n —l ) ( n —2 ) p ( p Jr a) (^ + 2 a) x{jx-f~ я) (я?“b 2a)

n (n —l) ( n — 2)p(pA~ a)(p-\-2a) ( l + a ) ( l + 2a)

lx=i

n (n —1)р{р-\-а)

1 + a

+ npj

, /о , n(n — l)p {p + a) np + (3 + a ) ---;--- ■ H---

x “ha x

n {n —l)p (p + «) + (3 + a) — --- +%P'

1 + a

]®=i

Corollary 3.1.

For the ordinary moments of a random variable with a binomial distribution function the following recurrence formula is true

(23) dar

ar+1 = npar{ n - l , p ) + p —^ -, where ar = ar(n, p ) .

Proof. Let ns pass to the limit in (21) as a -* 0, we have lim [npar (n—1, p + a , 1 + a) + (p/а) zip ar]

a—>0

np lim ^ Sir in ' -l)[*](p + a)[*’- a] , ..

a + «)*-*■ - +

^ ~ Г

— nP $i(n — l)^ p %Jr p i=0

dar

dp n par(n— 1, p) + p dar dp In the last member of this equality ar is already a moment in the binomial distribution.

Coro llary 3.2.

For the ordinary moments of a random variable with Poisson distribution the following recurrence formula is true

P roo f. Let in formula (23) n 00 and p -> 0, in such a manner that np — > X then

r r

n p ar(n—1, p) = np 81(п~1)[г]р г X У} SIP = Xar.

i = 0 i = 0

/

Putting np = X and taking into account that

we have

dar dar dX dar

dp dX dp dX

p dar{n, p)

dp np

dX

d (Id*

г = 0

The obtained formulae, in connection with (23) (obtained from (21))

prove formula (23).

1 3 8 W . D y c z k a

Co r o l l a r y 3 . 3 .

The ordinary moments for a negative binomial distri­

bution are expressed by the following recurrence formula ar+1 = (vlq)ar(v-j-l) — var, where ar ~ ar(v).

P ro o f. We evaluate the limit of the expression (21) under condi­

tions (g) of § 1.

We have

npar(n—1, p + a, 1 + a) = np 2 *

{ n - l p { p + a f > - a1

(l + a)[i’- a]

Я i~ 0 vp

q \ q L Aiar = 2 i

1 = 0

iSl

Е « ( ‘ + 1 Г ₀ - T 2 _{* i=0}

ar(— >

= — ar(v+l);

\ q I q

Г Г

^ iSjA t'’- 1"! = V ® (A )’ „[<.-■]

г=0 г=0 ' ® ^’

]h‘»-n]

r

= vA] ^ $ ( —j = rZlJar.

i=o ' ®'

For the (r+ l)-th moment of a random variable with a negative binomial distribution we thus obtain the formula

(25') ar+l = (vp/q)ar{v + l) — vA\ar

which after transformations takes form (25). In the above argument A\

means obviously the first difference in relation to v of the moment ar with the step 1.

The recurrence formula for the ordinary moments of the random variable connected with the urn scheme of Pólya can also be obtained from (21) if we assume there p = MJN and = sJN ; assuming further in that formula s — —1 we obtain the recurrence formula for a random variable with the hypergeometric distribution. The recurrence formulae for a random variable connected with the urn scheme of Pólya have been obtained by Sródka [8]. Formulae (23) and (24) are not new (e.g. [6]).

References

[1] H. C ra m e r, Metody matematyczne w statystyce, Warszawa 1958.

[2] W . F e lle r , Wstęp do rachunku prawdopodobieństwa, Warszawa 1966.

[3] M. F is z , E achun ek prawdopodobieństw a i statystyka matematyczna, Warszawa 1958.

W C. D . K e m p an d A . W . K e m p , Generalized hypergeometric distributions, Jo u rn a l of th e R o y a l S ta tistica l S o cie ty , Ser. В 18 (1 9 5 6 ), pp. 2 0 2 - 2 1 1 .

[5 ] M. Gr. K e n d a l l , The advanced theory of statistics I , L on d on 196 2 .

[6 ] J . R i o r d a n , Moment recurrence relations for binomial, Poisson and hypergeometric frequency distributions, A n n . M ath . S ta tie t. 7 (1 9 3 9 ).

[7 ] K . S a r k a d i , Generalized hypergeometric distributions. P u b licatio n s of S cience, volum e I I , fa sc. 1 - 2 , 1 9 6 7 .

[8 ] T . S r o d k a , Wzór rekurencyjny na momenty zwykle w rozkładzie Pólyi, P ra c e M at. 8 (1 9 6 4 ).

POLITECHNIKA ŁÓDZKA, K ATEDRA MATEMATYKI W YDZIAŁU CHEMICZNEGO