STA 3007 Applied Probability
2005
Tutorial 8
1.
Poisson Process
(a) Poisson Process
Definition
A Poisson process of intensity, or rate,
λ >
0
is an integervalued stochastic pro
cess
{
X
(
t
);
t
≥
0
}
for which
(1) for any time points
t
0
= 0
< t
1
< t
2
<
· · ·
< t
n
, the process increments
X
(
t
1
)

X
(
t
0
)
, X
(
t
2
)

X
(
t
1
)
, . . . , X
(
t
n
)

X
(
t
n

1
)
are independent random variables;
(2) for
s
≥
0
and
t >
0
, the random variable
X
(
s
+
t
)

X
(
s
)
has the Poisson distribution
Pr
{
X
(
s
+
t
)

X
(
s
) =
k
}
=
(
λt
)
k
e

λt
k
!
for k
= 0
,
1
, . . .
;
(3) X(0)=0.
In particular, observe that if
X
(
t
)
is a Poisson process of rate
λ >
0
, then the
moments are
E
[
X
(
t
)] =
λt
and
Var[
X
(
t
)] =
λt.
For
λ >
0
is a constant value, it is called a homogenous Poisson Process.
For
λ
=
λ
(
t
)
>
0
is a function of time, it is called a nonhomogenous Poisson Process.
Example:
i. Suppose that customers arrive at a facility according to a Poisson process
having rate
λ
= 2
. Let
X
(
t
)
be the number of customers that have arrived up
to time
t
. Determine the following probabilities and conditional probabilities:
(a)
Pr
{
X
(1) = 2
}
(b)
Pr
{
X
(1) = 2
and
X
(3) = 6
}
(c)
Pr
{
X
(1) = 2

X
(3) = 6
}
(d)
Pr
{
X
(3) = 6

X
(1) = 2
}
1
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ii. Let
{
X
(
t
);
t
≥
0
}
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 Spring '11
 KB
 Probability, Probability theory, Poisson process, two hours, Three hours, Siegbert

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