Denker
Spring 2010
416 Stochastic Modeling  Assignment 10
SOLUTIONS:
Problem 1:
(Problem 40, p. 352)
Let
N
1
and
N
2
be two independent Poisson processes. Show that
N
(
t
) =
N
1
(
t
)+
N
2
(
t
)
is a Poisson process and determine the rate of
N
.
Solution:
We show the properties of the Poisson process with rate
λ
1
+
λ
2
are satisFed.
1. Since
N
1
(0) =
N
2
(0) = 0 we also have
N
(0) =
N
1
(0) +
N
2
(0) = 0.
2. Let
s
i
< t
i
,
i
= 1
, .., n
, deFne
n
disjoint intervals, say
t
i
< s
i
+1
. Then
N
(
t
i
)

N
(
s
i
) =
N
1
(
t
i
)

N
1
(
s
i
) +
N
2
(
t
i
)

N
2
(
s
i
)
i
= 1
, ..., n.
Since all random variables
{
N
k
(
t
i
)

N
k
(
s
i
) :
k
= 1
,
2; 1
≤
i
≤
n
}
are independent
by assumption and the deFnition of a Poisson process, also the random variables
N
(
t
i
)

N
(
s
i
) are independent. Hence
N
(
t
) has independent increments.
3. Since
N
(
t
)

N
(
s
) is the sum of the two independent Poisson distributed random
variables
N
k
(
t
)

N
k
(
s
),
k
= 1
,
2, it is also Poisson distributed with rate being the
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 Spring '08
 DENKER
 Stochastic Modeling, Probability theory, Exponential distribution, Poisson process, insurance company, 20 minutes, one hour

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