IEOR 161 Operations Research II
University of California, Berkeley
Spring 2008
Homework 6 Suggested Solution
Chapter 5.
58. Let
X
j
be the time it takes for the first type j coupon collected.
P
{
i is the last type collected
}
=
P
{
X
i
=
max
j
=1
,...,n
X
j
}
=
∞
0
(
X
j
< x
∀
j
=
i
)
λ
i
e

λ
i
x
dx
=
∞
0
j
=
i
(1

e

λ
j
x
)
λ
i
e

λ
i
x
dx
=
1
0
j
=
i
(1

y

λ
j
/λ
i
)
dy
(
y
=
e

λ
i
x
)
=
E
j
=
i
(1

U
λ
j
/λ
i
)
68. (a)
E
[
A
(
t
)

N
(
t
) =
n
]
=
E
n
i
=1
A
i
e

α
(
t

S
i
)
=
E
[
A
]
e

αt
E
n
i
=1
e
αU
[
i
]
=
E
[
A
]
e

αt
E
n
i
=1
e
αU
i
=
nE
[
A
]
e

αt
E e
αU
=
nE
[
A
]
e

αt
t
0
e
αx
1
t
dx
=
nE
[
A
]
1

e

αt
αt
E
[
A
(
t
)] =
E
[
E
[
A
(
t
)

N
(
t
) =
n
]] =
E
[
N
(
t
)]
E
[
A
]
1

e

αt
αt
=
E
[
A
]
1

e

αt
αt
λt
=
E
[
A
]
λ
(1

e

αt
)
α
(b) Going backwards from time t to 0, events occur according to a Poisson process
and an event occurring at time s has value
Ae

αs
attached to it.
72. (a) Define the random variable
S
n
as the departure time of the last rider. Since it is
the sum of
n
independent exponentials with rate
λ
, it has a gamma distribution with
parameters
n
and
λ
.
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 Fall '08
 Lim
 Operations Research, Probability theory, Poisson process, independent Poisson process, Suggested Solution Chapter, independent exponentials

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