1
The Poisson Process
Let
!
>
0
.
A
Poisson process with parameter
distributes points along a line or
in
R
n
in a random fashion such that the distribution satisfies the following axioms.
(1)
Occurrences in disjoint intervals are independent.
(2)
In a measurable set with measure
A
, the probability of a single occurrence,
Pr(1,
A
)
, satisfies:
lim
A
!
0
Pr(1,
A
)
A
=
"
.
(3)
In a measurable set with measure
A
, the probability of
n
>
1
occurrences,
Pr(
n
,
A
)
, satisfies:
lim
A
!
0
Pr(
n
,
A
)
A
=
0
.
As derived in class, the exact probability of
n
occurrences in a measurable set of
measure
t
is exactly:
p
(
n
)
=
lim
k
"#
k
n
$
%
’
(
)
t
k
$
%
’
(
)
n
1
*
t
k
$
%
’
(
)
k
*
n
p
(
n
)
=
lim
k
"#
k
!
n
!(
n
$
k
)!
k
n
t
( )
n
1
$
t
k
%
’
(
)
*
k
1
$
t
k
%
’
(
)
*
$
n
p
(
n
)
=
lim
k
"#
k
k
k
$
1
k
!
k
$
n
+
1
k
%
’
(
)
*
(
t
)
n
n
!
%
’
(
)
*
1
$
t
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 Summer '08
 Kyung
 Statistics, Probability theory, #, $ 1, 1 $, measurable set

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