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ORIE 3510 – Homework 3 Solutions
Instructor: Mark E. Lewis
due 2PM, Wednesday February 15, 2012 (ORIE Hallway drop box)
1. (a)
{
X
n
}
is not a Markov chain. To see this, it sufces to check that
P
(
X
4
= 1

X
3
= 0
, X
2
= 1)
n
=
P
(
X
4
= 1

X
3
= 0
, X
2
=
−
1)
.
Indeed,
P
(
X
4
= 1

X
3
= 0
, X
2
= 1) =
P
(min(0
,
1) +
ǫ
4
= 1
, X
3
= 0
, X
2
= 1)
P
(
X
3
= 0
, X
2
= 1)
=
P
(
ǫ
4
= 1) =
1
2
,
but
P
(
X
4
= 1

X
3
= 0
, X
2
=
−
1) =
P
(min(0
,
−
1) +
ǫ
4
= 1
, X
3
= 0
, X
2
=
−
1)
P
(
X
3
= 0
, X
2
=
−
1)
=
P
(
ǫ
4
= 2) = 0
.
(b) De±ne
Y
n
= (
X
n
, X
n

1
),
n
= 1
,
2
, . . .
, then the sequence
Y
n
is a Markov chain. To prove
this, we see that
P
(
Y
n
= (
i
n
, i
n

1
)

Y
n

1
= (
i
n

1
, i
n

2
)
, . . . , Y
1
= (0
,
0))
=
P
(
X
n
=
i
n
, X
n

1
=
i
n

1

X
n

1
=
i
n

1
, X
n

2
=
i
n

2
, . . . , X
1
= 0
, X
0
= 0)
=
P
(min(
i
n

1
, i
n

2
) +
ǫ
n
=
i
n
, X
n

1
=
i
n

1
, . . . , X
0
= 0)
P
(
X
n

1
=
i
n

1
, . . . , X
0
= 0)
=
P
(
ǫ
n
=
i
n
−
min(
i
n

1
, i
n

2
))
=
P
(min(
i
n

1
, i
n

2
) +
ǫ
n
=
i
n
, X
n

1
=
i
n

1
, X
n

2
=
i
n

2
)
P
(
X
n

1
=
i
n

1
, X
n

2
=
i
n

2
)
=
P
(
Y
n
= (
i
n
, i
n

1
)

Y
n

1
= (
i
n

1
, i
n

2
))
.
2. We will show that
P
((
X
n
+1
, Y
n
+1
) = (
i
n
+1
, j
n
+1
)

(
X
n
, Y
n
) = (
i
n
, j
n
)
, . . . ,
(
X
0
, Y
0
) = (
i
0
, j
0
))
=
P
((
X
n
+1
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