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Homework Assignment 1
Queuing Theory: 56 645558 01
Problem 2  Sum of Random Variables
Tony Smaldone
[email protected]
http://www.sntco.com/tony/masters.html
July 26, 2001
Problem 2  Sum of Random Variables
Find the distribution (density) of
Y
=
X
1
+
X
2
for the two cases where a).
X
1
∼
Poisson
(
λ
1
)
and
X
2
∼
Poisson
(
λ
2
)
; and, b).
X
1
∼
Exp
(
λ
1
)
and
X
2
∼
Exp
(
λ
2
)
. Assume independence.
Solution
a). Poisson Distribution
Given two independent random variables
X
1
∼
Poisson
(
λ
1
)
and
X
2
∼
Poisson
(
λ
2
)
the question becomes what is the distribution of
Y
=
X
1
+
X
2
. Since
X
1
and
X
2
are independent events, the distribution of
Y
will be the sum of all possible
values of
X
1
and
X
2
which sum to
Y
. For example, if
P
[
Y
=
k
], then
P
[
Y
=
k
] =
P
[
X
1
= 0]
P
[
X
2
=
k
] +
P
[
X
1
= 1]
P
[
X
2
=
k

1] +
···
+
P
[
X
1
=
k
]
P
[
X
2
= 0]
=
k
X
n
=0
P
[
X
1
=
n
]
P
[
X
2
=
k

n
]
Expanding the summation
∑
k
n
=0
P
[
X
1
=
n
]
P
[
X
2
=
k

n
]
1
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This note was uploaded on 01/05/2010 for the course ECE 01 taught by Professor All during the Spring '09 term at Aarhus Universitet.
 Spring '09
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