VARIOUS PROPERTIES OF THE POISSON DISTRIBUTION
G. F. Hadley*
T. M. Whitin**
Consultants, The RAND Corporation
P-2421
August 25, 1961
*University of Chicago
**University of California, Berkeley
Any views expressed in this paper are those of the
author. They should not be interpreted as reflecting
the views of The RAND Corporation or the official
opinion or policy of any of its governmental or
private research sponsors. Papers are reproduced by
The RAND Corporation as a courtesy to members of its
staff.
P-2421
-1-
VARIOUS PROPERTIES OF THE
POISSON DISTRIBUTION*
G. F. Hadley
T. M. Whitin
Presented below is a table of properties of the Poisson distribution
which have been found useful in working with a variety of operational models.
In particular the properties have proved useful to the authors in
obtaining explicit expressions for inventory levels) backorders) and stockout costa under various assumptions concerning the lead time distribution
and penalty functions. Sone of the relationships are almost immediately
obvious) but they are included in this paper for the sake of completeness.
In the following table p(k) g), P(k) g) are defined as follows:
P(k, P) = rer e
; P(k, 4) ex 2: P(j, 4)
**
The relations given hold for all non-negative integers k) and for all g >0.
The gamma function of argument
a
is denotedbyT'(cx). L D(t)...7 represents
the Laplace Transform of f(t). L1 f(k) designates f(k) - f(k-1). (11.) repre.,
sents
212
I (n-r)!
LIST OF POISSON RELATIONSHIPS
1. k p(k) g)
it p(k -
1,
g)
2. kt p(k) g) gk p(0) g)
*
Some of these properties were derived under contract Nonr-2311(00) at
the Stanford Research Institute and were published without proof in
OpeNtions Research, May-June, 1961. The proofs are given here as well as
additional properties.
Although k is to be non-negative, it is possible in some of the properties to have a negative argument for p or P when k < 3. These properties
will be correct for such k) if the convention is used that p(j) g) 0,
p) 1. 1 for j a negative integer.
P-2421
-2-
E
11)P(k - j,
= P(k)
x)
J=0
03
E J PO,
j=k
5.
6.
= P0c. -
P(j) g) g
J=0
- P(k,
gg
l., J P(i, 4) = 4 g(k - 1, g) - P(x, pg
j=k
j=k
G2
010
02
(j - k)P(j) g) = E
3=0
J P(k + JP 4) =
2E: P(j, g) = g P(k - 1,
0 '. k P(k1g) mg
j=k+1
g p(k, g) + (g - k)P(k + 1, p)
8. k P(k + 1, g) = k P(k, g) g p(k - 1, g)
9. g P(k, g) - k P(k + 10 g) = g P(k - 1, g) k P(k, g)
to
g) = g P(k - 1, g) + (1 - k)P(k, g)
10.2::
j=k
12.
1E,
g) = g L14 (kl 47 — k P(k + 1, g) + 1
34)
p) = g P(k - 1, g) + 1 - k)P(k, g) - p P(x, g) + x P(x + 1, g)
13.
j=k
14. E J2 PO,
J -0
=
lt2
co
j=k
16.
J2 P(j, 4) = p. 1)(k - 1, 4) + 4211k - 2/ P)
k j2p(j, g) = p
3=0
- P(k,
r- P(k - 1,
+ g2 Ll
g17
P-2421
-3.
2
17. g: J PO,
j=k
= ff(k - 1, R) P(x, i47 + g2g(k - 2, g) - P(x - 1, 1.0.7
op
OD
18. 2E:
j=k
k)2PO,
p(j,
0
j- p(k + j, g) k2 P(k, g) + g(1 - 2k)P(k 1, g) +
j=0
42 F(k - 2, 4)
co
19. L.,j3 p(j,
j=k
g P(k - 3, g) + 3112 P(k - 2, 11) + g P(k - 1, g)
k
20.E j3 (j, 4) 43J P(k - 2, R17 + 342L1 (1=0
- 1, p17'+ pg. - Pot, pg
00
21.
j=0
j3 PO,4.
) = g3 3112 g
.g3g(k - 3, g) - P(x 2, gi7 + 3g2g(k - 2, g) P(x - 1, Kg+
j3 p(j,
22.
j=k
gg(k - 1, g) - P(x, p)7
op
23. 22
j=k
k)3P0, 1.4) =
E
j=0
,
JJ P(k J, 4) = k3 P(k, g)
g (1- 3k+ 3k2)P(k -1, g) + 3g2(1 - k)P(k -2, g) + g3 P(k - 3, 4)
00
r
in p(j, 0, n 0, 1, 2,
j=k
1
U-].
1, 2,
g
x(k-1
7:(7:
Then it(k) g ,
Th
^ )
x=0
24. Let gn(k)
25. Let 12n (k) =
jn P(j, p) 0 n = 0, 1, 2,
j=k
Then
_an (k)
(4)n+1-1 (n+1) /1
-1.110! (k)
n+1 f1=0
p1070. (k) (k_i)n+1 P(k,
n = 1, 2, ...
co
26.
2
(k) = 2 P(k-2, 4) 4 P(k-1, 4)
j P(j, 4) = I/1(k)
j=k
k(k-1) P(k, g)
2
P-24i1
ce
27.
.2
,
Pl.1, 14) =—+
2
2
28.
p(k - 1, Kg
P.) =
- p(k, p7 +
k(k
1)p(k
2
co
E
j PO,
O IL)
29.
j=k
Co
30.
PO/
j=k
2 /
%
E j Pk i, i.t )
-
P(J,
°1
.- -3- +-i- - ivP(k, 14)
= 1
j=tk
See (26)
2
+ LI P(k - 1, 14) +
P(k - 2, ti
1,3
'
3 P(k - 3: 60
o
00
E
32.
,2
P(J, P) I +
k
2
j o 14 ) =
3
2 + 3
k(2k + 3i(k +
1) p(k + 1, 14) +
P(k, 1417 +
j=0
Li _
2
P(k -
1, 1.0.7 E3
P(k 2, 111.7
P(x + 1, 4) +
j=k
.2.2
1-15.(k - 1,1.1) - P(x,
„3
3
34.
17 +
ff(k -
2, p,) - P(x - 1, 17 +
Zirqk - 3, p) - P(x - 2, 1.0
c114
m) - pot - 1, p) - p(k, II)
35 *
dp(k, KT) x 5(k 1, XT) - POE, x217
dT
36.
dP(IcItis 4) = p(k - 1, 14)
37.
dP(k, XT) = p(k dT
XT)
P-24-21
-5 -
38.
p(k,
39.
p(k,
xt)at
P(k
XT)
+
0
4o.
tn p(k, Xt)dt = 1 In
xn+1 k!
I
P(n + k + 1, XT), n Op 1, 20 ...
co
272 po, X.t= T P(k, XT) - P(k -I- 1, XT)
41. sp(k, Nt)dt
0
" j=k+1
4.2.
t
t) at = xn+1
P(k,
1
n+1
k(k ÷
2
p(k 4- 2,
co
S-1 (n
P(n
j+ 1
Tn+1
+ 1)
n)
p(k, ?
a)
P(n + k + 1, AT), n = 0, 1, 2, ...
If
(T - t)nP(k) 7■.t)dt =
E (- 1)r (rn)
03
f
r
P(k + r + 1, XT)..7
f(XT)1*113(k, XT)
a-1 -f3t
a
p(k) A„t)dt =
a
+
cc)
0
7
p
a-1 -tit
a
prk; 7,.(t + Ti7dt = E p(k j) T)B(j)
47.
4.8.
n)...7
2. (- 1).r (n)
Tri-r((r+
+ k))
(T - t)np(k) .,t)dt
7%
=)„
rP(r
k + 1) ,a)
r+1
x
rg)
14.
11.6.
XT
r ()2
2 P(k,
1
S
45.
1
t P(k, Xt)dt
eic)
ft
p
a-1 e-f3t
1
P(k, XidtdT =
5-‘ r- E Boy
r(a)
t)
0 0
B(k),
aop
>0
P-242I
-649.
JrOD j,t+T
P( Pt
o o
50.
p(k, Xt)
-pt
e
(a) Alt,
j=0
X p(k -
11 mr) 0-x(t-r)dr
,r
P(k -
-T)dt
Xi)) e -X(t
o
co
52. E aj PO, 11)
.1=0
4
53.E 40:" PO, 4)
J=0
ega
-
, cc any positive nuMber in
= ega ■ 1)
-
r )B(i)J
p(j -
-
. Jr0
51.P(k, ht) =
X
k
1
Properties
52-59
P(x + 1, 00_7
j P(i, P) = e°a 1)P (k, 014)
54.
j=k
55.
j=k
d
OD
P
g)
4
56. 2:: au PO, A
j=o
57.
j=k
e (a - 1) LE(k,
a ega a -
- P(x + 1, a 1)_7
1)
fee ep(cc-1) Pk,k,
aj P(j, 4) =
- „,k P(k,
)J
x
Ea
58.j=0
P(j, 4)= -a-qte e(a-1)Li - P(x
59.E 044
j=k
P(j, g) =
1 € Acr-1)Lfs(k,
c7-77160.
If
11100 =
- P(x + 1, al-9.7
1,
0417
+
ax4-1 P(x
+ 1, IA)
P(k, p) + ax+1 P(x + 1, 14)3
PO, g)
Then Agi(k) = - (k - 1) p(k 1, p) = g p(k - 2, g)
P-2421
61.If 111(k) =
j p(j„ g), /14,1(k) = k p(k, g) = g p(k-1, g)
jo
62.ify00
-k) p(j, g),AlIk) = - P(k, p)
j=k+1
63.Ift(k) = E (j-k) p(j, g),14/(k) = P(k, g) - 1
j=0
611.. L 5(k) N.t1/ -(54̀1'7 )k+1
65. L 5(k, Xt.17 = L 5(k-11 NtI7 L(Re-Xt)
k
66. L ff"(k, N.tv = --aop
67.Let 0n(k) =E jn agi p(j, g) 1 a any positive number in properties 67-84.
j=k
Then 0n(k) = QV
n-1 n-1)./
(k-1), n = 1, 2, ...
x
(
00
68. 2:: j a4
j=0
l4(a-1)
p(j) g) = 04 e
OD
69.
=k
j C114 POP g) = ag eil(la-1) P(k-11 ag)
k
j 00 p(j) g) = a4 e1) il - p(k)ok4)
70.
]
j=0
71. E j cx p(j, 4) = au el4a-1) [P(k-1, (14) P(x, ag)]
j=k
up
72. 2:: j2 04 P(j, 4) = (4 e(a-1) (1 4- ag)
i=0
P-21121
-8co
73.
E j2 aj po,
= au ev0-1)
(
P.k-1, c41)
+
“k-2,
j=k
k
2
4
74.E ja
p(j,
ag
110-1)
e
Q4
e&
- P(k, a4) +
[1
-
P(k-1, (111)j
j=0
x
0
4
75.
Z
=
p(J,
j=k
(a-1)
,
1:
Pkk-1,
+ 0
1-1 [P(k-20
P(x-10 atigj
oo
76.
E: p a p(j, g) = ag ega-1)
(0
44)2 4.
3ctil +
1
co
4
77.E J3 a)
PO
0eg1)
)
=
[k
2
.
apr p(t-31 aU)
+
3ag
P(k-2, 1:
114) + P(k -1, OIA
j=k
k
,
4
78. E JJ
g) =
ap
ega-1)&44)
2
- P(k-21 ag)] + 3aP [1 - P(k-1, 01.
4)]
+
J=0
1
79.
j3
p(j,
1
)
= 041 e
P(k,
(c2
.
)
1)
03.
[P(k-3,
-
P(x-2, 0
11)]
+
P(k-1,
+
j=k
3
041 [P(k-21 aP) - P(x-11
80.
Let
On(k)
n
a) P(j,
a
Then
en(k)
n
=
1:
2/
(k-i)'1
-
P(x, c4.3'
g)
,
I-
a4)
ak-1 P(k, g) + On(k)
+
n-1
-x
2::
x.3
exck]
0..
00
81.
i a)
PO,
g)
4. a2 il e
1) - oil eg(a-1)
0-1)'
j=0
82.
g
a
r, I- eP
= -
to
= Ej ai
(
k)
j=k
PO,
g)
(a-i)2
e
%
P(k, 0
42)
ak
(a-1)2
(a-kce+k)
P(k,
p)
+
a2
11
u(a-1)
p e'
P(k -1, al)
+
P-2421
-9-
83. 2:
J
a eg(a-1)
11
PO, g) = (a-I)2
ea 1)
ov)j + a24 a_i - [
1 - P(k, 001
a 2 [3. ak-1 0_10u+,
(a-1)
84.
± j po,
j=k
a
=
ega-1) [P(k, 04) - P(x+1, 04)]
(a-l)2
a2
g e4(a-1) [P(k-1, ag) - P(x, 04)]
(a-ka+k) P(k, 11)
(a-1)2
DERIVATIONS OF POISSON PROPERTIES
14 k p(k, g)
=
e-g p(k-1)
(k-1):
P. p(k-11 g)
24
k! p(k., g)= k! e
k
P.
= gk e-g
k f„
= P Ptv,
k
3.
Pkk+1, g)/
k -g
e 4
E p(il 4) P(k-j, X) = 2::
J=0 i!
41 0
X
° e (k-J)!
= e
Jo i•
k
e-61+70
(1-0-7)
k;
= p(k,, 1.14-2s.)
(k-i)
ax+1 (x..4zx+1) P (x+1, g)
(a-1)`
P-2421
-10cc,
4.
j=k
from (1)
J P(j, g) = it EPO-1, 4)
j=k
= g P(k-1, g)
5. E
cc
ao
p0, 4) °
j=o
EJ
PO, g)-
J=0
J PO, g)
j=k+1
from (4)
= g P(-1, 4) g P(k, 4)
= g
- P(k, g.)7 since P(-1) g) = 1
aD
X
6. j=k
E J P(j, g) j=k
E J P(j) g)
00
E J PO g)
j=x+1
= it P(k-1, g) - it P(x, g)
from (4)
Co
(J-k)p(j) 4) = t
a J P(jA-k, g)
7.
j=k
= p(k+1, g) + 2p(k+2, g) + 3p(k+3, 4) +
= p(k+1, g) + p(j+2, g) + p(k+3„ g) +
+ p(j+2, g) + p(k+3, g) +
+ p(k+3, g) +
aD
=
P(j, 4)
j=k+1
OD
00
(j-k)po,
j=k
=
j=k
J P(J, g) - k Z P(J, 4)
j=k
= g P(k-10 g) - k P(k, g)
= (g-k)P(k+1, g) + it p(k-11 g) + g p(k, g) k p(k„ g)
= g p(k) + (g-k)P(k+1, g)
8. k P(k+1, g) = k P(k, g) - k p(k, g)
k P(k, g) - g p(k-1„ g)
from (1)
from (1)
P-2421
-119. P(k, g)
k P(k+1, p) =g P(k, g)
k P(k„ g) + k p(k, p)
= g P(k, p) - k P(k, g) + g p(k-1, g) from (1)
= g P(k-1, p) - k P(k, g)
10. E p(j, g) = g P(k-2, g) + (1-k)P(k-11 p) from (7)
j=k
= g P(k-1, g) + (1-k)P(k, g) + g p(k-2, p) + (1-k)p(k-11 g)
= g P(k-1, g) + (1-k)P(k„ g)
OD
from (10)
11.z: p(j, 4) g + 1
j=0
12.E pol 4
CO
d,
E P(j, 4) - E P(j, 4)
j=k+1
j=0
T.-P(k, p)J - k P(k71-11 g) + 1 from (10)0(11)
= g
CO
x
Co
13.,. PO, g) = E Po, p.) - E Po, 4)
j=x+1
j=k
= g P(k-11 g) + (1-k)P(k, g) - g P(x, p) + x P(x+1, g)
do
14.E i2 P(i, 4) =
CO
j=0
i4 P(j-1, 4) from (1)
Co
Co
r (j-1)p(j-1, g) + g 2:: P(j-1, 4)
.Jos
OD
= g 2:p. p(j-21 g) + p from (1)
j=0
=
=4.1. +
Co
a, „,
15.Z: j` PO, 0 =IIEJP(j-1, 4) from (1)
j=k
j=k
Co
co
F, p(j-1, 4)
= 4 Z: (j-1)p(j-1, 4)
j=k
j=k
cd
= g2 t•—• p(j-2, p) + p P(k-1, g) from (1)
j=k
= p P(k-11 g) + g2 P(k-21 g)
P-2421
-1200
2
00
,
5:: i` PO,
16.z: j p(j, p
j=0
-
J=0
= g
7,
J2
po, g)
42 - 4 P(k, 4) - 42 P(k-1, 4) from (14)1 (15)
=
x2
17. 3 J PO,
j=k+1
,2 - P(k-1, g17
- P(k,
Cg 2
0 = e_, J PO, 0 - 2! J2 PO, 0
j=k
j=x+1
= g P(k-1, g ) + g2 P(k-2, g ) - p P(x, g ) - g 2 P(x-1, g )
.7 + g2 g(k-2, g) - P(x-1,
= p g(k-1, p) - P(x, 0
OD
00
18.2E:
j=k
(j-k)2
= g P(k-1,
po, g) =
0
j=k
19.
j=k
2k
72, PO, 0
j=k
n
e
OD
j=k
PO, 4)
2kg P(k-1, g ) + k2 P(k) 14 from (15), (4)
+ g2 P(k-2, p)
g ) + 4(1-2k)P(k-11 p) + R2 P(k-2, g)
00 ,
,
JJ
-
J` PO,
= k2 P(k,
op
CO
17
4) = g
0
j=k
from (1)
cir.)
=
=
g
(j+1)2
j=k-1
op
E:
j=k-1
po, 14
2J + 1)13(J, 14
= p3 P(k-3, g ) + 3g 2 P(k-2, g ) + g P(k-1, g) from (15),(4)
OD
010
20.
J3 PO, IA) =
j=0
1=0
j3 p(j, g) - 2:2 J3 po, g)
j=k+1
32 Li - P(k-1
= g3 5 - P(k-2, g17 +
op
21.
Li -
P(k, 4/7from (19)
1
5:: jJ p(j„ g ) = 43 + 112 + t from (19)
,
x
22.
1 14J +
j=k
j3 P(j, 4) =
j=k
J.) P(J) 4) -
OD
2:: J 3 PO, 4)
j=x+1
= g3 L( -3, g) - P(x-2, i47 +
g LT(k
342 g(k-2, p) - P(x-1, p)J
-1, g ) - P(x, 11/7 from (19)
P-2421
-13, 0
23. E O-k)3 PO, P) = E (J J- 3j" k
j=k
j=k
00
1)
.1(2- k3) PO, 0
= -k3 P(k, g) + g(1-3k + 3k2) P(k-1, g) + 3g2(1-k)P(k-2, p)
+ g3
00
142
221.
E
j=k
25.
OP
from (4), (15), (19)
P(k-3, g)
from (1)
...n, ein-1 p(k-1, g) n = 1, 2, ...
in P(j, 4) = 4 2..
j=k
00
= P E (+1)n-1 P(j, P.), n = 1, 2, ...
j=k-1
n-1
= 11 Er (a-)
x gx (k-1), n = 1, 2, ...
x=0
[j_i)n+1
jn+1]
p(j,
j=k
(k-1)n+1 P(k, g) .
x n+1 p(k, g) - (10-1)n+1 p(k+1„ g)
(k-1)n+1 P(k, g)
(a)
gn+1 (k)
E (0..3.)n+1
=k
aD n
=
j=k 1d0
z
in+1]
)n+1-1
j (n+1)
O P)
1 P,
(b) binomial expansion
Set (a) = (b) and solve for nth term of binomial expansion
OD
042 n 1 (..1)n+1-1 ji (111.1) pfj, g)
E jn p(i, g % = 1
%
2-1 1=0
) n+1 j=k
j=k
-
(k) - (k-1)11+1 P(k, g) , n = 1, 2, ...
+ gn+1
..)
. (k) +
2:: (...1)n+1-1 (n+1).0
1 fo.-1
/ n+1 1=0
to
4411
it)
(
.11+1 P(k, g) 1
- (k-1)
oo
n = 1, 2, ... [Def. of
jn P(j, g)]
(k) =
j=k
P-2421
-14-
26. t j p(j, 4) ...1/1 (k)
j=k
1
=1 nb (k) + p.2 (k) - (k-1)2 P(k, p.4
EP(k-1, g) + (1-k) P(k, p) + p. P(k-1, g) + p.2 P(k-2, g)
•
(k-1)2 P(k, p.)} from (10), (24), (15)
2
P(k-2, g) + g P(k-1,
2
40
27.
E
2
J P(J, g) =-+i
k(k-1) P(k,
2
from (26)
j=0
OD
00
k
J P(i, 4) - Z: J P(J, g)
28. rj J P(J) g) .
j=k+1
j=o
J=0
2
,
g)from (26),(27),
Li - P(k-1, g17 + p ff. - P(k, 417 4. k(:11-) Pkk+1,
OD
29.
j=k
J P(J,
J P(J,
=
j=k
- Z J P(J, 4)
i=x+1
aD
30. 2:: J` P(J, 0 =122 (k)
j=k
= f"(gPo OP1 113(k) - (k-1)3 P(k, 4)]fram (25)
p P(k -1, p)
=
(1-k)P(k, g) + 31';
2
P(k-2, g) +
3g P(k-1, g) - 3k(1;-1) P(k, 4) 43 P(k-3, 4)
3g2 P(k-2, g) + g P(k-1, g) - (k -1)3 P(k, 0.3from (10),
=
-
L
(26), (19)
k2 k
k3 ,
,- 7-
it3
P0c-3, 4)
3
P(k, p) + p P(k -1, g)
2 P(k -2, g)
P-2421
-150
OD
31. 2:: J`
j=0
g)
from (30)
g + 3'42
2 +3
o0
ao
32. 2:: J2 P(J) 4) =
J`P(J) g) - 272 J2 P(J) 4)
j=k+1
k(21(k+1) P(k+1, g) + g
2
2 Li - P(k-1, 147 + 3
do
j=k
J2 P(J) g)
j=k
0
J' P(J,
03
P(k, 147+
- P(k-21 g17 from (30), (31)
0
- 2:: J' P(J) 4)
j=x+1
-k(2k-1)(k-1) P(k, g) + x(21)
(x+1) P(x+1, g)
6
2
4 Lf(k-1, 4) - P(x,
.
3
3
34,
g(k-21 g) - P(x-1, 1117 +
g) - P(x-2,
e-P gk.
k!
k4k-1 e-11
14)
dg = k!
de(kl
= p(k-1, 4)
+
-
4)
dp(k, XT) k(XT)k-1 e-XTX X e-XT (XT)k
35. dT
k!
k!
= X 5(k-1, XT) - p(k, XT17
36.dP(k1 g) d
%
tk, 4) P(k+11 g) + ... 7
010 =;51 U
= p(k-1„ g) p(k, g) + p(k„ g) p(k+1 4) P(k+1, 4)
= p(k-1, g)
P(k+2,0+ ...from (34)
P-2421
-16-
37.
dT
NT) = d Azfu XT) + p(k+1, XT) +
dT 1- ‘4"
_7
X 5(k-1, NT) - p(k, NT) + p(k, NT) - p(k+1, NT) + ..._7 from (35)
= p(k-1., XT)
.0
38.
1
0415 +
f p(k) Nt)dt = - 5(k, Nt)_7
0
39.
f p k-1, Xt)dt (by parts)
0
p(k, Nt)dt = - k5(k,
+
p(k-11 Nt)dt (by parts)
T T
= -1
g p(k, NT) T. 5(k-1, X0_7) + s
=
r- NT) +
1 a(k,
X
=
1 a(k, NT) + p(k -1, XT) +
=
NT) +
+ p(1, NT)7 +
p(O,Nt)dt
+ p(1, NT) + p(0, NT) - iJ
X P(k+1, NT)
ho. ,
iF tn p(k, Nt)dt _ 1 (n+k)!
k!
Nt)dt
p(11+k, Xt)dt
1 (n+k)!
0+1 k! P(n+k +1, NT) from (39)
P-2421
-17-
P(k, Xt)dt =
14-1.
5(k, N.t) + p(k+1, Xt) +
1
(1+i,
L1 NT) + P(k+2, NT) 4-
from (39)
ro
1
E p(j, xr)
j=k+1
= T P(k, ).T) - P(k+1, NT)
_+2
42.
t P(k, Xt)dt
m2
= 2
T
P(k, M)7-
P(k) NT)
N. 1 (
2 3
T) 2 pot„
x2 L-( 2
n
t P(k, ht)dt =
43.
T2
1-
from (10)
p(k-1, Xt)dt
)P(k+2 xT)
from (40)
k(k-a) P(k+21 N.T)_7
2
f tn 5(k, Xt) + p(k+1, Xt) +
P( k+1, XT)
p(n+k+20 n)+
from (40)
1 57nLtlat P(n+j+11 xT)
n+1
A,
T
j=k
n+1
0
t- P(k, N.t)dt = n+1 P(k, Xt)3
Tn+.1 vof.t„
n+1 ""/
1
n+1
T tn+1 p(k-1, N.t)dt (by parts)
S
(n+k):
n+1
X (n+1)(k-1)
P(n+k+1„ XT)
P-2421
-18T n
jr
(-1)r (n
r) Tn-r tr p(k, Xt)dt
o r=0
E
44. f (T-t)n p(k, Xt)dt
. 2:
Tri-r
(-1)r TY
(r 'T
r=0
X
r (n) tr
45. f (T„on p(k, xt)dt =
0
rik+1, XT) from (40)
k!
•
J:
P(k, N.t)dt
(F43:7){(hT)
k
r=0
P(Icl 7,11)
P(k+r+1, NT)] from (43)
a-1 -pt
46
I PU/
e
p(pt)a-1 (kok e-(x+p)t
p(k, Nt)dt =
k+a-1 -(X*P)t
e
dt
t
e-u du
=
ri*ji 1 (X+PT
=B(k)
P-2421
-19-
pLE, X +TI7dt
47.
017 p(pt)a-1
t e -XT
k e
11(a)k!
X(t+T)
r044a-1 -(X413)t
Go
a k -XT
k
E
e
„Jo
a Xk e -XT k
k! r
cx
k -XT k
Xe
-XT E
j=0
dt
i rti+a_1 e t dt
o ro4c4)
(a+j)
(X+P) -(a+i) r(a+J)
r f a;
e
dt
X (1_ a
Nrq3) ( p+x)
(
B( j)
= E p(k-j, 7a) B(j)
3=0
,(pt)13-1 e-pt p(k) XT)dt dir
r (a)
48 .
(%t)3
ri
e-xt
dt from (39)
L 3=0
\k
( Z3/
= 1/x.
2:: B( J )]
J.0
la
X+f3 )
from (46)
P(Pt)a-1 et
J9.
r(a)
1
P(k) X rndt dfr
ay o u3.01U-1 e-pt
0
k
Nt+TA e-X(t+1
E
)rl(a)
dt
k
lht. [1
E
z p(j-i, AT) BM]
from (47)
j=0 i=0
50. ?,
p(k-1, hir) e-x(t-1-) 4
.
e
0
k e t
= (k-1):
t (4)k-1
di
(k-1)!
t
1k-1
dT
= p(k, xt)
51. P(k, At) . E p(k, At)
j=k
t
x E PO-1)
-X(t-in
o j=k
Jr
52.
e4:;
% p(t_il Al) e- X(t-1) dr
DOI+ a e p.
j=0
(Q14)2
+ •°•]
1. 2!
=
from (50)
P-2421
-21-
53.
fai
1 +
P(j, g) =
17 +
.1=0
e-g
=
E
faox
2
2. +
+ x!
,
p(j, ag)
J=0
= e
54.
MP
4
E
alj
j=k
1) [1 - P(x+1, ail)]
k+1
=
+ • 0 0
[1E2
AP
= ectile-g
•
2:: p(j, at)
j=k
P(k, QP)
4D
55.:!, aj PO,
j=k
=
j=k
PO, g) -
;E:
aj PO, P)
j=x+1
= eg(a-1){p(k, u4A) P(x+1„ all)] from (54)
a,
56.
E oi P(j, g) = P(0, )
p(1, g)
p(2,
J =0
+ a Lp(1, g) + p(2, g) + . •
1
2
%
+ a [p(2, g)
P(3, 11)
= p(01 11) + p(1„ g) a (1 + A) + p(2,
1
1+3. _
ai [(
a
jo
1
= 1-a
j=0
a e4a-1)- 1
a-1
=
42
E;!.:1
E aj
j=0
from (52)
PO, 0
a2(1 +
• •
0 0
L)
a2
4. • 0 •
P-2421
-22do
57.cz P(i, IA) =
ak [P(k,
j=k
+ p(k+1, g) + p(k+21 4) + ...]
+ ak+1 [p(k+1, g) + p(k+2, 4) +
a
...]
k+2 pkk+2, 4) +
...)
+ 0 0
ak p(k) 4) + aka
(1 + k p (k+1, g) ak+2 /1
1+ 1
-17;
a
40
J-1.1
E ccit+ilik) -
1.1 pze)
to
p(k+j, g)
go
E P(k+i'
J=0
=
p (k+2, g) +
+ c44-71
P(10-J, V)
J =0
= 1 (,„ g(a-1) P(k, aU) - ak P(k, g)] from (54)
c77
-1: e
x
58.
au P(J,
E
c° aj PO, 1L)
(xi P(j, ti)
j=x+1
J=0
.
_ P(x+1, 01-4)] + axa P(x+1, g) 3 from (56), (57)
e4(a_i)
co
co
j=k
j=x+1
59. E 04 po, g) = E: 04 P(j, g) j=k
1.
= Ex7i
60.
ell(a-1) [P(k,
(k) = E p(J,
j=k
04 P(j, P)
from
- P(x+1, MA)) - ak P(k, g) + ax+1 P(x+1, 11)1 (57),(58)
Ej p(J, P)
j=k-1
= p P(k -1, g) - 4 P(k-2, g) from (4)
= -g p(k -21 g)
= -(k-1) p(k-11 g) from (1)
P-2421
-23-
(k)
k-1
=E
E
i POI
p(JI
i=0
= p [1 - P(k, p)] p [1 - P(k-1, g)) from (5)
= p p(k-1, p)
from (1)
= k p(k„ g)
62.
AY(k) = L (j-k) PO, 0
-
E (j-k + 1) p(j, p)
j=k
j=k
= tc (i-k) P(j, 4) - et (i-k +1) PO 4)
j=k
j=k
=
gi, 4)
63.all ( k) = E
k-1
(i-k)
j=0
p(j, 4)
(j-k +
p(J, 11)
k-1
k-1
=E
(j-k +
-E
14)
(j-k)
1=0
k-1
E
p(J,
1=0
= P(k, p) - 1
sp e.(800t
s k at
(it)
64. L [p(k, Xt)] = f
k!
[(Xt)k e-(s+x)t ]
k! (s+X)
X
0 s+A.
=
s-1-24
k
(s+X)k+1
X
re-+)t dt
wo
e-(s+X)t (xt)(k-1)
(k-1):
at
P-2421
-2465 L
k-1
X()
X
(s+x)k s+x)
p(k-1, Xt) L(Xe-A.t
from (64)
xk
(s+x)k+1
= L Lp(k, XI)]
66.L
Xt)
= L [p(k Xt) + p(k+1, Xt) + ... 3
Xi
A (a+x) 1
.x+s (! k
S (sook+1.
)
1)
= 1 X
s s+4
67.0. (k) =
=
in 0') p(i 4)
j+k
E
(n 1)
13 (i -1, 4)
j=k
=Q1E
j=k-1
i+1) 04 PO, 0
00 n-1
= E E
(n3-cl) jx 0)
j=k-1 x=0
y)
,1,-,1)91x (k-1)
1
mo
68-79. Can be derived from (67). Only those involving E: will be given
In detail here, i.e., (69), (73), and (77).
j=k
P-2421
-2569. 01 (k) = a 00 (k-1)
from (67)
ega-1) P(k-11 Q4) from (54)
=
1
73. 02 (k) ° o4 E
x=0
= all e
2
M 0. (k-1) from (67)
1)
f
[Pkk-1,
,
E (;) ox
77. 03 (k)
+ 011 P(k-21 og)) from (69)0 (54)
(k-1) from (67)
x=0
= ap
[00 (k-1) + 201 (k-1) + 02 (k-1)1
= Qeti(a-1) [P(k-1, ag)
- (j-1)n
80.
j=k
P(k-2, ag)
ajA POI 4)
014)2
P(k-3, cg)]
(n ?- 1)
00
= .(k-1)n ak-1 r(k,
+
E jn
j=k
= -(k-l)n ak-1 P(k,
p(k)
(a)
r
j=k
00
=
°) - (j-1)n
to
14, n-1
in al P(i. P) - a
in °) P(J) P) '7"14•
(n) ix(-1)n-caj P0,4)
j=k
j=k
j=k x=0
P-2421
-26Setting (a) = (b) and solving for en (k)
en (k) =
81-84.
[- (k-1)
n-1
Z.
7
65 (1.31c)(-1)n-x ex (k)]
44 -1 P(k, g) + On (k) +
Obtainable from (80). Here only (82) will be given in detail. (81) is
special case of (82) for k = O.
82. el (k) = ac
—if
.. .[(1-k)
f
ok-1 P(k, g) + 01 (k) -A
en ( )1)
from (80)
=A7 [(1-k) 034-1 P(k, g) + OP e14-1) P(k-1, OP)]
1
a(a-1)
la eg(a-1) pot, ao . ak p t, 11)]
(
2
a
= - -----15
egia-1) P(k, mili 4..2._
0-1r
ak
4. _-___
(a..a+)
(a-1)2
4( ) .
P(k, g)
P°t-11 c4))
from (69), (57)