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Past exam of the mathematics course of the University of Cambridge
/
2023
/
iii
/
Paper 224
/
2
/
f
/
Solution
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Past exam of the mathematics course of the University of Cambridge
2023
iii
Paper 224
2
f
2026-09-28
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On
B
, part
b
gives
lo
g
2
P
n
=
lo
g
2
Q
n
−
lo
g
2
Q
n
(
B
)
. Hence the
information entropy
of the conditional
law
satisfies
H
(
X
1
n
)
=
−
∑
x
1
n
∈
B
P
n
(
x
1
n
)
lo
g
2
Q
n
(
x
1
n
)
+
lo
g
2
Q
n
(
B
)
.
(1)
Parts
d
and e imply
lo
g
2
Q
n
(
B
)
=
H
(
X
1
n
)
+
x
1
n
∈
B
∑
P
n
(
x
1
n
)
lo
g
2
Q
n
(
x
1
n
)
≤
n
H
(
P
)
+
n
a
∈
A
∑
P
(
a
)
lo
g
2
Q
(
a
)
=
−
n
D
(
P
∥
Q
)
.
(2)
Exponentiation
proves
Q
n
(
B
)
≤
2
−
n
D
(
P
∥
Q
)
.
Ancestors
(11)
f
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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