On the recurrence relation for the moments of inflated binomial and inflated Poisson distribution

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I : PRACE MATEMATYCZNE X Y II (1974)

L. S

(Lublin)

1 . I n t r o d u c t i o n .

The binomial and Poisson distributions are among- the most frequently used discrete distributions in applications of the- probability theory and statistics. The properties of these distributions,, and in particular, the recurrence relations of the moments of the men- tioned distributions were investigated by many authors. List of some of them may be found in [2].

Eecently M. P. Singh [5], Ж. S. Singh [6] and Pandey [4] have been discussing the so-called inflated binomial and inflated Poisson distribu­

tions. These inflated distributions were introduced for the situations which are described by simple (binomial or Poisson) distributions except for zero celles (or some others celles) which are inflated, that is, there are more observations than can be expected on the basis of simple (binomial or Poisson) distributions. The investigation of some recurrence relations for the moments of inflated binomial and inflated Poisson distribution seems to be useful.

The first part of this note considers the recurrence relations for the moments about the origin as well as the mean of inflated binomial and inflated Poisson distributions. In the second part, we give the recurrence relations between the incomplete and complete moments of the mentioned inflated distributions and the moments of their simple distributions.

2 . D e f i n i t i o n s a n d n o t a t i o n s .

The random variable X is said to have the inflated binomial distribution with parameters a, p , n, if the prob­

ability function of X is given by

On the recurrence relation for the moments of inflated binomial and inflated Poisson distribution

(1) p ( x ; a ,p ,n ) —P [ X = x~\ =

for x — l,

where 0 < a < 1, a + /? = 1 , 0 < p < l , p-{-q = 1.

The random variable X is said to have the inflated Poisson distribution with parameters a and A, if the probability function of X is given by

(

2

B + ae x —- Xх for æ = l, x l

ae~x—- Xх for x = 0, 1, ..., I —1, Z + 1 , ..., x i

where 0 < a < 1, a + /? = 1, A > 0.

Throughout this note the following notations will be used. mr - r -th moment of binomial or Poisson distribution, mr (s) — r -th incomplete moment of binomial or Poisson distribution, pr - r -th moment about the mean of binomial or Poisson distribution, pr (s) — r-th incomplete moment about the mean of binomial or Poisson distribution, ju* — r-th moment about the mean of the distribution (p-j-g)n_1. mr , mr (s), pr and jur(s) stand for the mentioned moments in the case of inflated binomial or Poisson distributions. E denotes the operations of raising the order of a moment by unity, i. e. Е[лг = pr+1.

3 . T h e r e c u r r e n c e r e l a t i o n s f o r t h e m o m e n t s o f i n f l a t e d b i n o m i a l d i s t r i b u t i o n .

First we shall prove

T

1. I f X is a random variable lav in g the inflated binomial distribution (1), then

r — 1

(3) mr = P p { l- n )(l + l ) r~l + (}qlr + p “ (fc + i)]^r-i-fe for r = 1, 2,

(4) [лг = (3lq(al — a n p f l — Pp(n — Z)(l + aZ — anp)r * —

for r = 1, 2,

— (anp + fil)pr_1 + np £ Г . l )^ j~

j=o ' J '

r —2 r - 2

~ P \ J / ^Я-i ~P{anp + fil) £ J J Pj

(5) mr+1

for r = 0, 1, 2, ... ;

filr (l — np) + npmr + pq dmr dp

(6)

ar p(l

np)r+1 — P(l

np )pr -\-pq ^anrp,

for r = 1, 2, ...

P roof. The characteristic function of the distribution (1) is given by

<p(t) = № и + а{рей -\-а)п.

Writing it = в and putting

we obtain

0j __

3 = 0

2

—— — mj = ($1евг + anpee(pee + q)n 1

- pien - p npe6(-l+^ npee v i dj Pe + q pe + q jL j j

v i 2 71"»- Hence

4 ^ 01 % ■ » [)J

(pee-\-q) У —--- -Шл = ftl(pee + q)eei — finpee{l+1) + прев У —

2-/ (i — l) :

L

\

3 = 1 3 = 0 J

ej m'ji

and further

00 00

dj -

< 7 >

k

j

' j= l 2

D 7

ek(i + i ) k oi

ekik

6k+j

k= 0 k = 0 k= 0 j= 0

Using the similar consideration for the random variables X — EX , where EX, in this case, denotes the mathematical expection of X , we

have ✓

<’> 1

^ 1

Z

w

oo oo

/с— 1 3=1 dj j

U - 1)! h

00 00 (атгр + pi) — fij p (anp + /?ï) J ÿ J ^

03+k

1 = 0

0 0 OO

&=i i=o * !j!

rr^ j +

\ i \7 6j+k _

+ я р 2 2

т ш ^ +

/с

3 = о

fc ДА;

V i {ai — anp ) и v v + / % 2 ; - — ^ r — + ^ ( z- * ) 2 ;

/c = 0

* = 0

(1 + al — anpy 6

id

By (7) and (8) after the identification coefficients in 0r_1, we obtain (3) and (4) respectively.

Writing mr and pr as

(9)

( 10 ⁾

mr

Hr

n n

Y k rP [ X = fc] = ftlr + a Y ¥

k~o fc=o

n

£ ( k - E X ) P [ X = k]

k =о

n

ft(al — anp)r -f a ^ (k — anp — ftl)r p kqn~k

*== о ' '

n 'p kqn~k,

and differentiating (9) and (10) with respect to p, we get

and

dmr dp

П

7 - ^ ]

k =o J

n

4 ( r 4 “2 >+10 f V i - _{F4- k—0}

~ T a K + ° È V ^

= ----mr+ x — npmr + ftlr(np — l ) , pq

dpr - 1 ft (l — Tip ) — ar ft(l — np)r+l

= — a m ' H r - i + -— H r+1 "I--- - V r ---

dp pq pq pq

which give (5) and (6).

It is easy to verify that for a = 1 and Z = 0, formulas (3), (4), (5) and f6) pass into the well-known formulas for the r-th moments of binom­

ial distribution.

4 . T h e r e c u r r e n c e r e l a t i o n s f o r t h e m o m e n t s o f i n f l a t e d P o i s s o n d i s t r i b u t i o n .

The recurrence relations for the moments of this distribution, лее can find on the basis Theorem 1 and the following

L

1. I f n-> oo and p -> 0 so that \imnp X > 0 is fulfilled ,

then the probability function of the inflated binomial distribution (1) tends to the probability function of the inflated Poisson distribution (2) with p ara­

meters a and 1 .

The proof of this Lemma is analogous to the proof of the Theorem

of Poisson. Moreover, it can be observed that Lemma 1 can take a more

general form which would constitue an extension of the results of

Krysieki [3]. We need here the result of Lemma 1 only.

T

2. I f X is a random variable having an inflated Poisson distribution (2), then

r - l . .

(11) mr ^ p r - m i + i y - ' + X ^ П K - i - * fc=o ' 1 for r = 1, 2, ...;

(12)

r - l f i r = (31 ( a l

—

1 — /Щ 1—

—

^ j

k=Q ' ' for r = 1, 2

;

( ^{Ô/YYh \ mr -\ — jjj~ \}

for r = 1, 2, ... ;

(14) Я +i = arP(l — X)r+1 — P ( l —X)pr + x{arpr_

+ for r = 1, 2, ...

The proof of this Theorem is obvious.

In the case a = 1 and 1 = 0, we get the well-known recurrence relations for the moments of Poisson distribution.

5. The relations between the moments of inflated binomial and inflated Poisson distributions and their simple distributions. There are many the recurrence formulas for the moments of simple binomial and Poisson distributions. Just for that reason an investigation recurrence relations between the moments of inflated binomial and inflated Poisson distribu­

tions and the moments of their simple distributions may be useful.

The establishment of the recurrence relations "*for the incomplete and complete moments about origin of the inflated binomial and the inflated Poisson distributions does not make any trouble since

mr = /ЗГ + amr foi* r = 1, 2, ..., _ /йг + amr (s) for s < l, r = 1, 2, ..., mr (s) =

amr (s) for s > l , r = 1, 2, ...

The case of the recurrence relations for the incomplete and complete moments about the mean of these distributions is not the same as the above. To find these relations, we need the following obvious

L

2. I f X is a random variable having the r-th moment, then

(15) £ (» ) = %

'where pr(s) denotes the r-th incomplete moment about the point c, and pr(s)

the r-th moment about the mean EX.

In the cases of binomial and Poisson distributions we have respec­

tively (16) and (17)

T

^(*) = Д ^ )(ю # -с )г“^ ( в ) ,

r

/u^s) = 2 (rh - c y - j M * )

3 = 0 w

for s > 0 and r = 1 , 2 , . . . Lemma 2 allows to prove

T

3. I f X is a random variable having an inflated binomial distribution (1), then

Г 3 — 2

A + ap ^ P r~j { n p - l) r- j [nq £ ^ T 1) ^ 8) -

(18) pr (s) =

3 = 1 г=0

3 - 2

f or s > 1’

i= 0 ' ^

Г 3 — 2

B + ap £ f t pr- j ( n p - l ) r~j [nq

3 = 1 ' i= 0 ' '

- for S < 1

г=0 ' '

for r = 1, 2, 3, ...;

Г

pra^np - l ) rp0{s) + <mp £ Ir. j j8r_i(wp - Z)r_i X

X [(-Я + ff)i - V Î ( « - l ) - ^ _ i ( s ) ] for s > l,

T

a ^ n p - i y i ^ p ^ + i - i y a ^ + a n p x

X F 4 (np - l ) r- j [(E + q)j ~l p l (

- 1 ) - ^ {s)] for s < l for r = 1, 2, 3, ... ;

г 3' — 1

A + < m p q 2 2 (i) ( L i r ’ in p - W - H E - p Ÿ - ' - ' t i i s ) 3

for S > l,

r j — 1

5 + - я | | ( J l f T 1) r H n p - i r ' w - p ^ - ' p U » ) for s < l (19) pr (s)

(20) pr (e) =

for r = 1, 2, 3,

(

21

A + a 2 {•} F 4 (nP - l Y4 1(1 + x i=2 ' 3 '

x(npq/u0(s ) - p iu 1{s)) for s > l , Г

Б + а ^ ( ! ) р - Ц п р - г у ^ К г + Е у - ' - Е 3- 1] x j t i '

x(»Mi«o(*)-№(*)| f or S < 1 for r = 1, 2, 3, ... ; wüere

J. = af}r (np — l)r/

,0(

) + asqP(s)(s — np)~l [{s — an p — fïl)r — {3r (np — l)r]>

В = arp(l — np)r jr aftr (np — l)r [/Li0(s )—sqP (s)(s — np)~1] +

-\-asqP(s)(s — np)~1(s — anp — ;

(22) Я - С + а Д 1 (j) Рг~ *(пр-1)г- * [ п р 2 £ P i~ P £ (j 7 1) i“f+i]

or r = 1, 2, ... ;

T

(23) ft. = G + a n p g f y +

for r = 1, 2, 3, ... ;

Г ^ — 1

(24) ft, = C + a»M ( j ) ^ 1) Г Ч п р - г Г Ч Е - р ) 1- ' - 1? :

for r = 1, 2, 3, ...;

T

(25) f t = c + a W ) ^ ( « р - г г ' к г + ^ у - ' - ^ ' - ' К я и л - т ) /or r = 1, 2, ..., wftere

(7 = a / ? ( Z - w p ) r [ a f - 1 - ( - j0 ) r - 1 ] .

P roo f. We shall prove (18) and (22) only. The proofs of the another relations are similar.

By Frisch [1] formula for the r -th incomplete moment of the binomial distribution

r- 2 r- 2

M *) = в 2 Р ( в ) ( в - ^ Г “ 1 + а д ^ , Г “ 1)/|1.(* )-^ £ Г Т /

i =0 ' * ' '

lem m a 2 and some elementary calculations, we obtain for s > l

T

Я 00 = a ? ( n p - l ) rfi0(s) + a Pr~}(np - 1)гЧ pj(s)

T

= o / r(« f-i)> l(» ) + « 2 '( 5 1 P'~l (np - Z)r_i[sïP(s) (s - np)r l +

+»psj£ ( T 1) От1) л

(«)]

i=0 ' 7 i=0 ' ' J

= afir(np — l)r p 0(s)-\-asqP(s)(s — np)~l [(s — anp — ($l)r —

j* j _2 У"”2

-/ r< »i> -*n+ ai> w ) ( T 1) л - 2 1 f T V + i l

j=l ' 7 L i=0 ' 7 i=0 ' ' J

and for s < l

r

Я(«) = ar^(2-wjp)r + a^r(wj)-Z)riM0(s) + a ^ ,^J pr~j ( n p - l ) r~j pj(s)

= ar fi(l — np)r + a(ir(n p — l)r [/LlQ(s) — sqP(s)(s — np)~1] + Jг asqP(s) (s — np)~1(s — anp — fil)r +

+ aî> i^(') ^ - , ( « р - г г , [*ч<У ( T 1) f t - j ! О т 1) ft+>]

J=1 i— 0

which proves (18). Putting s = 0 in this formula we get (22).

The recurrence relations between the moments of inflated Poisson

■distribution and simple Poisson distribution gives

T

4. I f X is a random variable having the inflated Poisson distribution (2), then

<26) pr (s)

fo r r = 1, 2,

<27)

( А + аХ2У(*)(} for s>l,

J j = l г = 0 ^•7 / ' 7

U +аЯ £ 2 (fil’-f1) Рча-1У-‘М») for

\ j = i i =0 ' ' 7

■ 5

а/3 '( Я - г ) > (,(«) + а Я у , ( ^ ^ ( Я - г Г ^ ( Я + 1 )'-1Х

X /^o (^ 1) f^j—i (^0] for

r

ар (1

{Г

J = 1 W/

x[(Æ/ + l ) i - Vo(« — l) - ^ -- i( « ) ] /о»*

for r = 1, 2, 3,

P - ' V - V n - i - i W (28) M O = ’ f 7-1

for s > 1,

for s < l

for r = 1, 2, 3, ... ;

(29) fir {s)

А + а Х ^ ^ Г Ч ь - г Г 1 [{1 + щ ‘ - ' - & - ч м * )

3 = 2 '

for s > Z, Г

В + а Л 2 ( ,Л Г * 0 - 1 ) ' - , \.0 + Е )1- 1- Е 1- Ч М * )

3 = 2 X

for s < l for r = 1 , 2 , . . . , w&oro

A = a^(A-Z)r//0(s) + a5P (s)(s-A )-1[ ( s - a A - ^ ) r - ^ ( A - Z ) r], P = ar/S(ï-A)r + ajSr(A-Z)r [iM0( e ) - e P ( e ) ( s - A r 1] +

+ asP(s) (в - A)"1

(30) Я = C + a l (Г) (y7 ‘) r H l - i r ' l H

3 = 1 i= 0 w ’ K 1

=0+«я JÆ* (I) (V)

p-’v-vr1*-!-*

/or r = 1 , 2 , . . . ;

r

(31) Я = C + aA y ^ j ^ - ' O - Z r ' K l + JS)'-1- ^ - 1]/.,,

j=2 '

for r = 1, 2, 3 , . . . , гоЛеге

C = aj5(?-A )r [ar- 1- ( - / S ) r- 1].

The assertion of this theorem follows from the Theorem 3 and Lemma 1.

In the applications, the case in which an inflated there is at the point

^ = 0, is particulary important. The recurrence relations for the moments of the considered distributions in this case we can obtain from the corresponding formulas given above. The explicite formulas are long and we have omitted them here.

11 — Koczniki PTM — Prace Matematyczne XVII.

References

[1] R. F risch ,

Biom. 17 (1925), p. 165-171.

[2] T. G e rste rn k o rn ,

a b ility d istrib u tio n s,

Dies. Math. 83 (1971), p. 1-45.

[3] W. K ry s ic k i,

Bull. Soc. Sci. de Lôdz (8) 7 (1957), p. 1-24.

[4] K. N. P a n d ey,

J. Sci. Res. Banares Hindu Univ. 15 (2) (1964-1965).

[5] M. P. Singh,

J. Sci. Res. Banares Hindu Univ.

16, 1 (1965-1966), p. 87-90.

[6] S. N. Singh,

J. Indian Stat. Assoc. 1 (3)

(1963), p. 140-144.