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TABLES OF POISSON RANDOM NUMBERS
by
Peter A. Lachenbruch
University of North Carolina
at Chapel Hill
Institute of Statistics Mimeo Series No. 532
May, 1967
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DEPARTMENT OF BIOSTATISTICS
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UNIVERS ITY OF NORTH CAROLINA
Chapel Hill, N. C.
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Tables of Poisson Random Numbers
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The Poisson distribution is used in many fields.
Familiar
examples of its application are radioactive decay, emission of electrons
from vacuum tubes, telephone service, and traffic problems.
It is also
used widely in the field of stochastic processes, the Poisson process
being of fundamental importance.
The Poisson distribution may be found as a limiting case of the
binomial or from basic assumptions.
We shall briefly outline the deriva-
tion.
Suppose the probability of an event occurring in a time interval
~t
is approximately
in time
~t +
is (1 -
~~t).
0
as
~~t
~t +
(~ >
O.
0) and that the chance of more than one event
Thus, the chance of no event occurring in
n
Consider a time interval T which is divided into
subintervals of length
~t =
Tin.
The chance that exactly
k
n-k
~T
n )
•
This can be written
T)n-k
( 1 - .l!...!..
n
P(k)
n(n-1) ••• (n-k+1)
=
As n
+
00
nk
PT)n-k
.
(1 - - n
(since we want the intervals to tend to 0), this becomes
P(k)
equal
events occur
in time T is given by the binomial
_
~t
I
-2-
For a unit interval of time, we have
Prob (k events) = e
-]1
k
k:
]l
The mean of the Poisson Distribution is
co
l:
k=O
ke-]1 }l k
k:
e-]1 )1 k-l
(k-l) :
co
=
l:
]1
k=l
.
]1
Its variance is
co
co
e
l:
-]1
]l
k-2
]1.
(k-2):
k=O
Higher moments can be found using the moment generating function:
co
M(t)
tx
I
e
t x
Ix:
e
-]1
]1
x
Ix:
x=O
co
(]1e )
=
An important property that we shall use here is that the sum of
two independent Poisson variables with parameters ]11 and ]12 has a Poisson
11
,I
distribution with parameter ]11 + ]12·
Proof:
Let
x 2 - Pois (]12)
x 1 - Pois (]1l)
e
f(x ,x ) =
l 2
-]11
xl
]11
x
-]12
e
]12
,
,
x2'
xl·
-(]1l+]12)
e
]11
,
x
xl
]12
,
xl· x 2 •
2
II
1
Iij
2
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11
... ~
-Ii
,i
1
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-3-
w
Let
v
t
t
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X
z
.
w-v
-(].11+].1Z)
e
f(w,v)
=
].11
v
].1Z
(w-v): v:
w-v
v
W].1l
].12
~ (w-v): v: •
v=O
f(w)
v~
=
>1
a-
= Xz
Then
t
I
xl +
By using this property repeatedly, we can generate Poisson random
numbers of any multiple of our original parameters.
Thus, to find a Poisson
random number with parameter .4, we could add together four Poisson random
numbers with parameter .1.
The technique used for generating the random numbers is known as
the Probability Integral transformation.
If X is a random number from some
distribution with cumulative distribution F(x), then F(x) has a uniform (0,1)
distribution.
Thus, there is a one-to-one correspondence between the uniform
distribution and the distribution with cumulative F(x).
On the following pages are tables of Poisson distributed random
variables.
They were generated on the IBM 360/30 at UNC.
The statistics
of the uniform random number generator showed that it gave frequencies in
each of the intervals .00(.01)1.00 not significantly different (x 2 l00
= 93.5).
Serial correlations of lags 1,Z,3,4,5 were likewise not significantly
different from O.
These statistics are based on samples of size 10000.
The statistics for the Poisson variables are as follows.
test of variance was used.
The X2
I
-4-
]1
=
]1
= .50
2
X
2
X
.10
= 10210.2
= 9760.0
]1
]1
X2
= 10.0
]1
X2
X2
= 1.0
= 5.0
= 9845.0
= 9859.4
= 9894.3
Counts
=
]1
X
n
.10
=
]1
.50
= 1.0
]1
]1
E(n)
n
E(n)
n
E(n)
n
= 5.0
= 10.0
]1
E(n)
E(n)
n
0
9030
9048
6156
6065
3688
3679
58
67
1
0
1
921
905
2972
3033
3706
3679
302
337
5
5
2
47
45
738
758
1826
1839
809
842
30
23
3
2
2
120
126
595
613
1427
1404
69
76
4
0
0
12
16
148
153
1822
1755
184
189
5
2
2
34
31
1723
1755
376
378
6
0
0
3
5
1506
1462
669
631
0
1
979
1044
906
901
0
693
653
1088
1126
9
371
363
1297
1251
10
168
181
1231
1251
11
83
82
1145
1137
12
42
34
931
948
13
8
13
763
729
14
6
5
503
521
15
1
2
329
347
16
1
0
214
217
17
1
119
128
18
0
76
71
19
34
37
20
18
19
21
7
9
22
3
4
23
1
2
24
0
1
7
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25
0
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-5TABLE OF 10000 POISSON RANDOM VARIATES WITH PARAMETER = 0.10
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-38TABLE OF 10000 POISSON RANDOM VARIATES WITH PARAMETER
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3
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2
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3
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6
6
5
2
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5
4
4
3
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6
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3
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2
4
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2
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1
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2
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3
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9
3
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6
5
3
6
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2
5
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3
3
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3
4
2
3
0
6
5
8
5 6
3 10
6
3
7
3
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5
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6
2
2
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3
7
4
3
2
7
3
7
4
6
6
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6
3
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2
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6
4
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3
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2
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3
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1
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3
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5
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6
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4
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9
6
4
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= 5.00
3 10
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6
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5
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6
3
7
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6
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4
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5
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4
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3
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7
1
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