A Compound of an Inflated Pascal Distribution with the Poisson One

A C T A U N I V E R S I T A T I S L O D Z I E N S I S

F O L IA O E C O N O M IC A 194, 2005

ľadeusz Gerstenkorn*

A C O M P O U N D OF AN INFLATED PASCAL D ISTR IBU TIO N W ITH T H E P O ISS O N O N E

Abstract

In this p a p e r there is presented a c o m p o u n d o f an inflated Pascal d istrib u tio n with the P oisson one.

In the in tro d u c to ry p a rt o f the p a p er is giving an overview o f the last resu lts in topic o f co m p o u n d in g o f d istrib u tio n s, considering also the Polish resu lts. In succeeding Sections, p ro b ab ility function o f the com pound distrib u tio n Pascal-Poisson, factorial, crude and incom plete m o m en ts as well recu rren ce re la tio n s o f th is d istrib u tio n are presen ted .

M S C Iassilication: 60

Key words: c o m p o u n d d istrib u tio n s, c o m p o u n d inflated Pascal-P oisson d istrib u tio n , inflated Pascal d istrib u tio n , P oisson d istrib u tio n , facto rial, c rude and incom plete m o m en ts, recurrence re la tio n s for the m om ents.

I. IN T R O D U C T IO N

T h e problem o f the com pounding o f probability d istrib u tio n s goes back to the tw enties o f the 20lh century. It is w orth here to m en tio n the papers by M . G reenw ood and G .U . Y ule (1920; they to o k th e p aram eter o f the Poisson d istrib u tio n as a gam m a variate) and E.S. P earson (1925; he gave a device o f m aking the param eter o f a distributio n o f Bayes theorem ). In forties the problem was dealt w ith by L. L und berg (1940; the Pólya d istrib u tio n as a com pound one), F .E . S atterth w aite (see a rem ark on the com p ariso n o f S atterth w aite’s idea o f the generalized P oisson d istribu tion w ith a com p o u n d distribution in W. F eller’s paper, 1943; 390), W. Feller (1943; som e com pou nd d istributions as “ co n tag io u s” ones), G . Skellam (1948; the binom ial-beta giving Pólya-Eggenberger distribution), M .E. C ansado

* P ro fesso r, F a c u lty o f M ath em atics, U niversity o f Ł ó d ź and U niversity o f T ra d e , Ł ódź.

(1948; com poun d Poisson distributions). Som e in terrelatio n s am o ng com ­ p o u n d and generalized d istribution s gave J. G urlan d (1957). Im p o rtan t theoretical problem s arc contained in papers by II. T cicher (I960, 1962), W. M o len aar (1965) and W. M o lenaar and W .R . Zw ct (1966). A lso in sixties we find the results by T. Ś ro d k a (1964; R ayleigh with gam m a, gam m a with R ayleigh; 1966 - Laplace with n o rm al N (0, 3) left trun cated at zero, with gam m a, R ayleigh, W cibull, exponential), by S.K. K a tti and J. G u rla n d (1967; Poisson-Pascal), D.S. D ubcy (1968; com p o u n d W cibull). T h en we have p apers by D. S. D ubcy (1970; co m p ou nd gam m a, beta and F ), T. G e rsten k o rn and Ś rodka (1972; generalized gam m a with gam m a), H . Jakuszcnkow (1973; Poisson with norm al N(m, n) left trun cated at zero, w ith generalized tw o -p a ram eter gam m a, with M axw ell an d R ayleigh), W. D yczka (1973; binom ial with beta (incom plete m om ents)) T. G ersten ­ k orn (1982; binom ial with generalized beta), T. G e rsten k o rn (1994; gene­ ralized gam m a with exponential), T. G ersten k o rn (1996; Pólya with beta), G .E . W ilm ot (1989; Poisson-Pascal and others). M .L . H u an g and K .Y . b u n g (1993; D com pound Poisson d istributio n as a new extension o f the N eym an T ype A distribution).

A wide list of references to the problem considered on e can find in review studies by N .L. Joh n so n and S. K o tz (1969; ch a p te r 8: 183-215), G. P. Patil et al. (1968) and now adays by G. W im m er and G . A ltm ann (1999).

II. T H E C O M P O U N D IN G O F P R O B A B IL IT Y D IS T R IB U T IO N S

Below, we give a definition and relations needed for the com pounding o f d istribu tions.

Definition 1. T he com pound distribution

L et a ra n d o m variable X y have a d istributio n function ,Г(л:|у) depending on a p aram eter y. Suppose th a t the param eter у is considered as a random variable Y with a distribution function G(y). T h en the d istrib u tio n having the d istrib u tio n function o f X defined by

H (x ) =

J

— 00

will be called co m pound, with th a t с is an arb itra ry co n stan t o r a con stan t bounded to som e interval (G urland, 1957; 265).

I he occurrcncc of the co n stan t с in (1) has a practical justification because the d istrib u tio n o f a rand om variable, describing a p henom enon,

often depends on the param eter which is a realization o f an o th er ran do m variable m ultiplied, how ever, by a certain constan t.

T h e variable which has distribution function (1) will be w ritten dow n in sym bol as “X л Y " and called a coi. .pound o f the variable X with respect to the “ co m p o u n d in g ” У. R elation (1) will be w ritten sym bolically as follows:

H (x ) = F (x \c y ) A G ( y ). (2)

У

C o n sid er the case w hen b o th variables are discrete, the first one with a probab ility fu nction P (X = X i \kn) depending on a p aram eter n being a ra n d o m variable N with a probability function P (N = n).

T h en (1) is expressed by the form ula

h(Xl) = P ( X = x t) = £ P ( X = x t\N = k n ) - P ( N = n). (3) /1=0

III. T H E IN F L A T E D PA S C A L л P O IS S O N Y

Definition 2. T h e distrib u tio n of the form

P£X = i\Y) = 1 — s + s ■ P ( X = j'ol ľ ) for i = i0,

s • P (X = i\Y ) for i = 0, 1, 2, i0 - l , i 0 + l , n,

(4)

where 0 < s < 1 and the variable У, being in the condition, has a distribution

G (y), we call an inflated conditional distribution. It is a discrete d istrib utio n

w ith an inflation o f the distrib u tio n P (X = i|Y ) at a point i = i0.

D istribu tion (4) were introduced, nam ed as inflated, by S.N. Singh (1963) and M .P . Singh (1965-66, 1966) and later discussed by m any au th o rs. Some in fo rm atio n a b o u t these results one can find in T. G e rsten k o rn (1977) and G. W im m er, G. A ltm an n (1999).

Lemma 1. T h e com pounding o f an inflated d istrib u tio n w ith an o th er d istrib u tio n gives a distrib u tio n with the same inflation.

T h e p ro o f, as ra th e r easy one, is here om itted.

Theorem 1. W e assum e th a t X is a v ariab le w ith in flated Pascal d istrib u tio n P a s{nk, p) depending on a p aram eter n, i.e.

Pas(nk, p) - Pa,( X = i\N = n) =

Cnkf°•"«

1 №>•-». °

Let N be a variable with the Poisson distrib u tio n Pa(n, Я)

P0( N = n, Я) = — e ~ A, n = 0, 1, 2, ..., Я > 0 .

n!

By using the form ula (3) we get

M O =

1 — s + s P - ; ‘olexp(0 — Я) for i = i0,

h'-for i = 0, 1, ..., i0 - 1, i0 + 1, ... (3)

w here p > О, Я > 0, 0 = qk and

Vm = Í * [i’ l]d F (x) (6)

is the so-called ascending (or reverse) factorial m o m en t determ ined by the equality:

(Г

Mm = Е[(кХУ ‘-~4 = X ( k n f ‘- exp( - 0),

n\

Proof. A t first we determ ine /1,(10) = P (X = i0): hs(i0) = P S(X = i0) = 00 rni-V'-- 1 ! J = £ ( l - s + s ----7 7— p ioqnk) - cxp( — Я) = n= 0 ■ 1 «о! п=о n\ (7)

A nalogously we calculate /1,(1').

If in (5) we take s = 1 we obtain the com pound Pascal-Poisson distribution w ith o u t inflation

h(i) = - - ^ ‘- е х р ( 0 - Л ) , / = 0, 1, 2, ...

R. Shum w ay and J. G urlan d (1960) have also o b tain ed this result but using a m ethod o f generalization o f distributions.

T h e m o m en t (6) or (7) is to be determ ined in dependence on crude o r factorial m om ent.

In the p ap e r by R. Shum w ay and J. G u rla n d (1960; 92) we find the follow ing form ula

м и = z s : k ‘ f s ‘j

/= i ;= i

w here Sj are Stirling num bers o f the second kind, к is a p aram eter and <*[,■) is a factorial m o m ent

“m = I x tadF (x) — 00

where x r‘] = x ( x — l)(x — 2 )...(x — i + 1) and S ' arc Stirling num bers o f the th ird kind.

In general the factorial m om ent a m are easier to be calculated th a n the crude ones. But one can pass from the factorial m o m ents to the crude ones.

T his problem has been exactly discussed in the p ap e r by T. G ersten k o rn (1983) where one can also find a table o f Stirling num b ers o f th e third kind.

IV . T H E C O M P O U N D P A S C A L -P O IS S O N D IS T R IB U T IO N A N D IT S M O M E N T S

T he com pound Pascal-Poisson w ithout inflation can be written symbolically

Pap(k, p, Я) ~ Pa(nk, p) л P 0(X)

П

T his d istrib u tio n has been presented in an equivalent form by S.K. K a tti and J. G u rla n d (1961) as a generalization o f the Poisson distribution w ith the P ascal one.

Corollary 1. T h e m o m en t o f the rlh order o f the co m p o u n d d istribution P ascal-P oisson is the m ean value o f an a p p ro p ria te m o m en t o f Pascal d istrib u tio n if the param eter n appearing in the form ulae o f m om ents o f the Pascal d istrib u tio n is considering as a ran d o m variable o f the Poisson d istrib u tio n , i. e.

fyr](k, P, X) = a[r](nk, p) л P 0( l )

П

br(k, p, A) = ar(nk, p) л P 0(A).

П

T ak in g in account these form ulae, we have

Theorem 2. T he factorial m om ent b[r](k, p, A) o f the com pound distribution P ascal-Poisson is expressed by

b[r](k, p, A) = u fá (8)

where ^ir] is the reverse factorial m om ent o f the Poisson d istrib u tio n , i.e.

t w !r' - n - , e x P( - X ) . (9)

n = 0 n '

Proof. b[r](k,p, A) = л В Д = ( ^ j ( ( n k ) [r~ 1] л Р Д ) ) =

A lready in G. B ohlm ann (1913; 398) we find the re latio n between the crude and factorial m om ents

br = Í s rjb lr] (10)

]= о

w here S j are Stirling num bers o f the second kind. T he table o f these n u m b e rs ca n be fo u n d , fo r in stan ce, in A. K a u fm a n n (1968) o r in J. Ł ukaszew icz and M. W arm us (1956).

Corollary 2. D irectly from (8) and (10) we have

br( k ,p , A) = W - Y / ť l (11)

j= o W

Rccurrcncc relations for crudc m oments

Theorem 3. C rude relations o f the com pound P ascal-P oisson Pap(k, p, A) d istrib u tio n arc expressed by

1) br+ t (k, p, X) = £ ( ) £ S}bu+ n(/c, p, A), (12) i = o V / J = 0 2) br+ |(fc, p, A) = Yj S fr v + n(k, P, A) -I- - ^-br(k, p, A). (13) 4 0 P ro of A d 1)

br+ i(k , p , A) = ar+ i(n/c, p) л P„(A) =

/I

= nkP„ í ( - V m + 1. p) a =

<11=0 V1/ "

= f Ź ( r\ n k a f r k + l , p ) A P„(А)). (14)

4 i-0 \ v я

Now we will calculate the expression in parentheses using (10), some properties o f factorial polynom ials, and the following proposition:

Proposition 1. F actorial m om en t o f the Pascal Pa(n k ,p ) d istrib u tio n is given by W ' ’ 11 (see: D yczka, 1973: 223 (63), (64)). n ka ^n k + 1, p) л P„(A) = П = I S lj ( - \ \ n k + l ) ^ - i ] л P 0(A) = 7=0 W = Ż S j^ Y (n fc(n fc + l))W --4 A P 0(A) = ;=o \ 9 / я = Z S j ^ W +1' - 1 , A P o(A )= j=o \ 9 / n ^=0

A d 2) W. D yczka (1973: 225 (70)) has show n th a t

a,+ i(n k, p) = P (nkar (nk + l , p ) + — / . ar(n k ,p )).

4

J p

A

Using this fo rm u la in (14) and regarding the first co m p o n en t, we have

nk ^ ar(nk + 1, p) л Pn{X) = nk ^ £ S ' ( J ) (nk + 1 ) [J-~ 11 л Pn( l ) = J

J

=

is'jblj+

1](k,p,k).

J = 0 W j = o

I hen regarding the second co m ponent we have

P d ar(nk, p) л P0(X) =

J

f „/pV

n

A n incom plete (truncated, generalized) crude m om en t mr(t) o f o rd er r o f a d istrib u tio n function F (x) is defined by

00

mr(t) = jV d F (x ).

F o r incom plete factorial m om en t we have analogously mrn(0 = \ x [r]dF (x).

In the discrete case we assum e th a t t in the sum is an integer. It is quite evident th at a com plete m o m ent is a special case o f the incom plete one.

W. D y czka (1973: 212, (22)) has show n th a t incom plete cru de m om ents arc expressed by the incom plete factorial ones in th e follow ing way:

*=o

w here S', are Stirling num bers of the second kind. U sing this n o tio n for our case, we have

Theorem 4. Incom plete factorial m om ents o f the com pound Pascal-Poisson

Pap(k, p, A) d istrib u tio n are expressed by

where 0 = qkl and ß [r + 4 is the reverse factorial m o m en t given by (9). Proof. W . D yczka (1973: 227, (73)) has show n th a t

delink, p, t) = a[r](nk, p)a0(nk + r , p , t - r ) = i - о

C o m p o u n d in g this expression w ith P 0(A), we have

(<f*(n/c)tr+,,- 1]) л P 0(;.)) = A.

П

L et us calculate the expression in parantheses:

00 (ak) У

qnk(nkyr+i.-i] л p до = у (nfc)[,+1 .-1 1 H - ^ - e x p ( - Я) =

„=n n!

= exp(0 - A) £ (nk)[r+l- 11 exp( - 0) = exp(0 - ?)ц{г+п.

„ = r. n\

00 nn

T ak in g in account these calculations, we have A = t-r-lp T + l u w - E t_{1= n} exp(° ~ ^ + = = br(k, p, A) - exp(0 - X)pr £ P.. ß [r+n-I = П »•

As a sim ple conclusion, we have

C orollary 3. Incom plete crude m om ents o f the co m p o u n d Pascal-Poisson

Pap(k, p, A) d istrib u tio n are expressed by

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Tadeusz Gerstenkorn

ZŁOŻENIE INFLACYJNEGO ROZKŁADU PASCALA / . ROZKŁADEM POISSONA

Streszczenie

W p racy p rezen to w an e jest złożenie inflacyjnego ro zk ład u Pascala z ro zk ład em Poissona. W części w stępnej p racy p o d a n y je s t przegląd w yników badaw czych d otyczących tem atu złożeń ro zk ład ó w ze szczególnym uwzględnieniem polskich a u to ró w .

W d alszych rozd ziałach p o d a n o funkcję p raw d o p o d o b ień stw a ro zk ład u złożonego Pascal- -Poisson o ra z jeg o m o m en ty silniow e, zwykłe, niek o m p letn e o raz zw iązki rekurencyjne.