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Past exam of the mathematics course of the University of Cambridge
/
2023
/
iii
/
Paper 224
/
2
/
a
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Mathematics course of the University of Cambridge
Past exam of the mathematics course of the University of Cambridge
2023
iii
Paper 224
2
2026-09-28
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Table of contents
Solution
a
Solution
0
0
0
a
The
type
of
x
1
n
∈
A
n
is its
empirical distribution
P
x
1
n
(
a
)
=
n
1
∑
i
=
1
n
1
{
x
i
=
a
}
,
a
∈
A
.
(1)
Its
type class
is
T
(
P
)
=
{
y
1
n
∈
A
n
:
P
y
1
n
=
P
}
.
(2)
Every string in
T
(
P
)
has
Q
n
-
probability
∏
a
∈
A
Q
(
a
)
n
P
(
a
)
=
2
−
n
{
H
(
P
)
+
D
(
P
∥
Q
)}
,
(3)
while the
method of types
gives
∣
T
(
P
)
∣
≤
2
n
H
(
P
)
. Therefore
Q
n
(
T
(
P
))
≤
2
−
n
D
(
P
∥
Q
)
.
(4)
Ancestors
(10)
2
Paper 224
iii
2023
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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