MS&E 221
Problem Set 1 Solutions
CA: Hongsong Yuan
January 19, 2011
Problem 1
(a)
E
[
X
n
] = 0
(b)
Var[
X
] = Var[
E
[
X

P
]] +
E
[Var[
X

P
]]
Var[
X
n
]
=
Var[
E
[
X
n

X
n

1
]] +
E
[Var[
X
n

X
n

1
]]]
=
0 +
E
[
βX
2
n

1
]
=
β
E
[
X
2
n

1
]
=
β
(Var[
X
n

1
] + (
E
[
X
n

1
])
2
)
=
β
Var[
X
n

1
]
Var[
X
1
]
=
βx
2
0
Var[
X
n
]
=
β
n
x
2
0
Note: The sequence is not necessarily a Markov chain. The current mean and variance just depend
on the previous step, but the distribution itself could depend on all previous steps.
Problem 2
Let
Ω
be the probability space and
A, B, C
be 3 disjoint subsets that evenly divides
Ω
, that is,
P
(
A
) =
P
(
B
) =
P
(
C
) =
1
3
. Define
X, Y, Z
as follows:
X
=

2
I
(
A
) +
I
(
B
) +
I
(
C
)
Y
=
0
Z
=

I
(
A
)

I
(
B
) + 2
I
(
C
)
where
I
(
·
)
is the indicator function. Then,
P
(
X > Y
)
=
P
(
B
) +
P
(
C
) =
2
3
P
(
Y > Z
)
=
P
(
A
) +
P
(
B
) =
2
3
P
(
Z > X
)
=
P
(
C
) +
P
(
A
) =
2
3
Problem 3
For each part, I’ll give both the reason for modeling the process as a Markov chain
and the reason for not doing so.
1
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(a) Yes: in a simple but widely used model, one assumes that the increase of a stock price from
today to tomorrow consists of a fixed increasing rate plus a fluctuated increasing rate, i.e.
S
t
+1

S
t
=
S
t
(
μ
t
+
Z
t
)
, where
μ
t
are constants and
Z
t
are independent (possibly identically
distributed) through
t
.
No: in many cases, historical data of the stock prices contributes largely in predicting the
future price. For example, if the stock price has been falling over the past few weeks, then
probably for tomorrow we cannot expect a high rate of return as usual.
(b) Yes: when a new bidder arrives, the current bid is the highest among all previous bids. One
would naturally assume that the new bid is the current bid plus an increment that is indepen
dent of the bidding history.
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 Winter '11
 Ramesh
 Probability theory, Markov chain, PCC PT C PGC PAC PT C PAC

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