solution of homework 2
Problem 1
Let
V
n
(
x
) represent the maximum expected value function before the
nth
bet
given the fortune before the
nth
bet is
x
. So we have the following equations:
V
n
(
x
) = max
0
≤
α
≤
1
[
p
n
V
n
+1
(
x
+
αx
) + (1

p
n
)
V
n
+1
(
x

αx
)]
V
N
+1
(
x
) = ln(
x
)
(1)
Based on the above equations, we could get:
V
N
(
x
) = max
0
≤
α
≤
1
[
p
N
ln(
x
+
αx
) + (1

p
N
) ln(
x

αx
)]
= max
0
≤
α
≤
1
[
p
N
ln(1 +
α
) + (1

p
N
) ln(1

α
)] + ln(
x
)
The optimal value for
α
is
α
*
N
= (2
p
N

1)
+
and
V
N
(
x
)
*
=
C
*
N
+ ln(
x
).
C
*
N
=
p
N
ln(1 +
α
*
N
) + (1

p
N
) ln(1

α
*
N
)
By induction, if
V
n
(
x
) =
C
+ ln(
x
),
C
is constant. Then through equation
1, we could get:
α
*
n

1
= (2
p
n

1

1)
+
(2)
C
*
n

1
=
p
n

1
ln(1 +
α
*
n

1
) + (1

p
n

1
) ln(1

α
*
n

1
)
(3)
V
n

1
(
x
) =
C
*
n

1
+
C
+ ln(
x
) =
C
*
n

1
+
V
n
(
x
)
(4)
Therefore, for any 1
≤
n
≤
N
, we have:
α
*
n
= (2
p
n

1)
+
(5)
C
*
n
=
p
n
ln(1 +
α
*
n
) + (1

p
n
) ln(1

α
*
n
)
(6)
V
n
(
x
) =
N
X
i
=
n
C
*
i
+ ln(
x
)
(7)
The above is the strategy of gambling.
Problem 2
Let
V
n
(
x, p
) represent the maximum expected value function before the
nth
bet
given the fortune before the
nth
bet is
x
and the probability of winning is
p
. So
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 Spring '12
 Richard
 Probability theory, Englishlanguage films, Nikon, Nikon Fmount, Nikon F5, αn

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