ľadeusz Gerstenkorn*

ACTA
UNIVERSITATIS
LODZIENSIS
F O L IA O E C O N O M IC A 194, 2005
ľadeusz Gerstenkorn*
A C O M PO U N D OF AN INFLATED PASCAL DISTRIBUTIO N
W ITH TH E PO ISSO N O NE
Abstract
In this p a p er there is presented a com pound o f an inflated Pascal d istrib u tio n with the
Poisson one.
In the in tro d u c to ry p a rt o f the pap er is giving an overview o f the last results 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 results. In succeeding Sections,
probability function o f the com pound distribution Pascal-Poisson, factorial, crude and incom plete
m om ents as well recurrence relations o f this distrib u tio n are presented.
M S C Iassilication: 60
Key words: com pound distributions, com pound inflated Pascal-P oisson d istrib u tio n , inflated
Pascal d istrib u tio n , Poisson distribution, factorial, crude and incom plete m om ents, recurrence
relatio 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 distributions goes back
to the tw enties o f the 20lh century. It is w orth here to m ention the papers
by M . G reenw ood and G .U . Yule (1920; they to o k the p aram eter o f the
Poisson d istrib u tio n as a gam m a variate) and E.S. Pearson (1925; he gave
a device o f m aking the param eter o f a distribution o f Bayes theorem ). In
forties the problem was dealt with by L. Lundberg (1940; the Pólya
d istribution as a com pound one), F.E. S atterthw aite (see a rem ark on the
com parison o f S atterthw aite’s idea o f the generalized Poisson distribution
w ith a com pound distribution in W. Feller’s paper, 1943; 390), W. Feller
(1943; some com pound distributions as “ contagious” ones), G . Skellam
(1948; the binom ial-beta giving Pólya-Eggenberger distribution), M .E. Cansado
* Professor, F a c u lty o f M athem atics, U niversity o f Ł ódź and U niversity o f T rad e , Ł ódź.
[ 21 ]
(1948; com pound Poisson distributions). Some interrelations am ong com ­
p ound and generalized distributions gave J. G urland (1957). Im portant
theoretical problem s arc contained in papers by II. Tcicher (I960, 1962),
W. M o lenaar (1965) and W. M olenaar and W .R. Zwct (1966). Also in
sixties we find the results by T. Ś rodka (1964; R ayleigh with gam m a,
gam m a with Rayleigh; 1966 - Laplace with norm al N(0, 3) left truncated at
zero, with gam m a, Rayleigh, Wcibull, exponential), by S.K. K atti and
J. G u rlan d (1967; Poisson-Pascal), D.S. D ubcy (1968; com pound W cibull).
T h en we have papers by D. S. D ubcy (1970; com pound gam m a, beta and
F ), T. G ersten k orn and Środka (1972; generalized gam m a with gam m a),
H. Jakuszcnkow (1973; Poisson with norm al N(m, n) left truncated at zero,
with generalized tw o-param eter gam m a, with M axwell and Rayleigh),
W. D yczka (1973; binom ial with beta (incom plete m om ents)) T. G ersten­
korn (1982; binom ial with generalized beta), T. G erstenkorn (1994; gene­
ralized gam m a with exponential), T. G erstenkorn (1996; Pólya with beta),
G .E . W ilm ot (1989; Poisson-Pascal and others). M .L . H uang and K.Y.
b u n g (1993; D com pound Poisson distribution as a new extension o f the
N eym an T ype A distribution).
A wide list of references to the problem considered one can find in
review studies by N.L. Johnson and S. K otz (1969; chapter 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 PR 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 distributions.
Definition 1. T he com pound distribution
L et a ran d o m variable X y have a distribution function ,Г(л:|у) depending
on a p aram eter y. Suppose th at the param eter у is considered as a random
variable Y with a distribution function G(y). T hen the distribution having
the distribution function o f X defined by
H (x) =
J
F (x\cy)dG (y)
—00
(1)
will be called com pound, with th a t с is an arbitrary constant o r a constant
bounded to som e interval (G urland, 1957; 265).
I he occurrcncc of the constant с in (1) has a practical justification
because the d istribution o f a random variable, describing a phenom enon,
often depends on the param eter which is a realization o f an o th er random
variable m ultiplied, however, by a certain constant.
T h e variable which has distribution function (1) will be w ritten down
in sym bol as “X л Y " and called a coi. .pound o f the variable X with
respect to the “ co m p ounding” У. R elation (1) will be w ritten symbolically
as follows:
H (x) = F (x \c y ) A G (y ).
(2)
У
C onsider the case when both variables are discrete, the first one with
a probability function P (X = X i \kn) depending on a p aram eter n being
a ran d o m variable N with a probability function P (N = n).
T hen (1) is expressed by the form ula
h(Xl) = P (X = x t) = £ P ( X = x t\N = k n ) -P (N = n).
/1=0
III. T H E IN F L A T E D PA SC A L л
(3)
P O IS S O N
Y
Definition 2. T he distribution 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 distribution
with an inflation o f the distribution P (X = i|Y ) at a point i = i0.
D istribution (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 authors. Some
inform ation ab o u t these results one can find in T. G erstenkorn (1977) and
G. W im m er, G. A ltm ann (1999).
Lemma 1. T he com pounding o f an inflated distrib u tio n with another
distrib u tio n gives a distribution with the same inflation.
T h e proof, as rather easy one, is here om itted.
Theorem 1. W e assum e th a t X is a variable w ith inflated Pascal
d istrib u tio n P as{nk, p) depending on a p aram eter n, i.e.
Pas(nk, p) - Pa,( X = i\N = n) =
1
—s + s
Cnkf°•"«
,0 *
------------n
(-----p ° qnk for i = i0
i«!
1iS—
№>•-».
°nk
j — p-q
where p >
0
, /c >
0
, p+q=
for i =
1
0
,
1, 2 ,
i0 ~
1,
i0 +
1,
...
and
x 11' - 11 = jc(x + l)(x + 2)...(x + 1 - 1)
is the so-called ascending factorial polynom e.
Let N be a variable with the Poisson distribution Pa(n, Я)
P0(N = n, Я) = — e~ A,
n!
n = 0, 1, 2, ...,
Я>0.
By using the form ula (3) we get
MO =
1 — s + s P - ; ‘olexp(0 — Я) for i = i0,
h'for i =
(3)
0 , 1,
..., i0 -
1,
i0 +
1,
...
w here p > О, Я > 0, 0 = qk and
Vm =
(6)
Í * [i’ l]dF (x)
is the so-called ascending (or reverse) factorial m om ent determ ined by the
equality:
Mm = Е[(кХУ ‘-~4 = X ( k n f ‘-
(Г
n\
exp( - 0),
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 ithout inflation
h(i) = - - ^ ‘-е х р ( 0 - Л ) ,
/ = 0, 1, 2, ...
R. Shum w ay and J. G urland (1960) have also obtained 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 or
factorial m om ent.
In the p ap er by R. Shumway and J. G urland (1960; 92) we find the
follow ing form ula
м и = z s : k ‘ f s ‘j
/=i
;= i
where Sj are Stirling num bers o f the second kind, к is a param eter and
<*[,■) is a factorial m om 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
third kind.
In general the factorial m om ent a m are easier to be calculated th an the
crude ones. But one can pass from the factorial m om ents to the crude ones.
T his problem has been exactly discussed in the paper by T. G erstenkorn
(1983) where one can also find a table o f Stirling num bers o f the 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 without inflation can be written symbolically
Pap(k, p, Я) ~ Pa(nk, p) л P0(X)
П
T his d istribution has been presented in an equivalent form by S.K.
K atti and J. G u rlan d (1961) as a generalization o f the Poisson distribution
w ith the Pascal one.
Corollary 1. T he m om ent o f the rlh order o f the com pound distribution
Pascal-Poisson is the m ean value o f an ap p ro p riate m om ent o f Pascal
d istribution if the param eter n appearing in the form ulae o f m om ents of
the Pascal distribution is considering as a random 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
Pascal-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 distribution, i.e.
t w
n= 0
Proof. b[r](k,p, A) =
!r' - n - , e x P( - X ) .
(9)
n'
л В Д = ( ^ j( ( n k ) [r~1] л Р Д )) =
A lready in G. B ohlm ann (1913; 398) we find the relation between the
crude and factorial m om ents
br = Í s rjb lr]
]= о
(10)
where S j are Stirling num bers o f the second kind. T he table o f these
n u m b ers can be fo u n d , fo r instance, in A. K a u fm a n n (1968) o r in
J. Łukaszew 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
j= o
W
where S rj are Stirling num bers o f the second kind.
(11)
Rccurrcncc relations for crudc moments
Theorem 3. C rude relations o f the com pound Pascal-Poisson Pap(k, p, A)
distribution 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) = Y j S fr v + n(k, P, A) -I- -
^-br(k, p, A).
(13)
4 0
P roof
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)).
4 i-0 \ v
я
(14)
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 ent o f the Pascal Pa(n k ,p ) distribution is
given by
W
' ’ 11
(see: D yczka, 1973: 223 (63), (64)).
nka^nk + 1, p) л P„(A) =
П
=
I S lj ( - \ \ n k + l ) ^ - i ] л P 0(A) =
7=0 W
= Ż S j^Y (nfc(nfc + l))W --4 A P 0(A) =
;=o \ 9 /
я
= Z S j ^ W
j=o \ 9 /
+1' - 1 , A P o(A )=
n
^=0
T ak in g account this result and (11) in (13) we have (12).
Ad 2) W. D yczka (1973: 225 (70)) has show n th at
a,+ i(nk, p) = P (nkar (nk + l ,p ) + — / . ar(n k ,p )).
4
J p
A
Using this form ula in (14) and regarding the first com ponent, we have
J
nk ^ ar(nk + 1, p) л Pn{X) = nk ^ £ S' ( J ) (nk + 1)[J-~ 11 л Pn( l ) =
J
=
t ^ ^ Y +V ü b t , =
J= 0
W
is'jblj+1](k,p,k).
j =o
I hen regarding the second com ponent we have
P
d
ar(nk, p) л P0(X) =
J
d{
P
f „/pV
d
P
d
u
<l
n
T he result (13) is then evident.
Incomplete moments
An incom plete (truncated, generalized) crude m om ent mr(t) o f order r o f
a distrib u tio n function F(x) is defined by
00
mr(t) = jV d F (x ).
t
F o r incom plete factorial m om ent we have analogously
mrn(0 = \ x [r]dF (x).
In the discrete case we assum e th at t in the sum is an integer. It is
quite evident th at a com plete m om ent is a special case o f the incom plete one.
W. D yczka (1973: 212, (22)) has shown th at incom plete crude m om ents
arc expressed by the incom plete factorial ones in the follow ing way:
*=o
where S', are Stirling num bers of the second kind.
U sing this no 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) distribution are expressed by
i- о
where 0 = qkl and ß [r +4 is the reverse factorial m om ent given by (9).
Proof. W. D yczka (1973: 227, (73)) has shown th at
delink, p, t) = a[r](nk, p)a0(nk + r , p , t - r ) =
C om 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
Let 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!
00
nn
+
n11ľ . exp( - 0) = exp(0 - ?)ц{г+п.
= exp(0 - A) £V (nk)[r+l„ = r.
n\
T ak in g in account these calculations, we have
t-r-lp T + l
A =
uw -
En t
1=
exp(° ~ ^
+ =
= br(k, p, A) - exp(0 - X)pr £ P.. ß [r+nI=П »•
As a sim ple conclusion, we have
C orollary 3. Incom plete crude m om ents o f the com pound Pascal-Poisson
Pap(k, p, A) d istribution are expressed by
r
hr( k ,p ,X ,t) = X ( S ;/) M( / c ,p ,A ) - p j c x p ( f l- ;.) L
цИи+ц-
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Tadeusz Gerstenkorn
ZŁOŻENIE INFLACYJNEGO ROZKŁADU PASCALA
/ . ROZKŁADEM POISSONA
Streszczenie
W pracy prezentow ane jest złożenie inflacyjnego rozkładu Pascala z rozkładem Poissona.
W części wstępnej pracy p o dany jest przegląd w yników badaw czych dotyczących tem atu
złożeń rozkładów ze szczególnym uwzględnieniem polskich a u torów .
W dalszych rozdziałach p o d a n o funkcję p raw dopodobieństw a rozkładu złożonego Pascal-Poisson oraz jeg o m om enty silniowe, zwykłe, niekom pletne oraz związki rekurencyjne.