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Past exam of the mathematics course of the University of Cambridge
/
2025
/
iii
/
Paper 164
/
1
/
ii
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Mathematics course of the University of Cambridge
Past exam of the mathematics course of the University of Cambridge
2025
iii
Paper 164
1
Created
2026-09-24
Updated
2026-09-24
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Solution
ii
Solution
0
0
0
ii
Write
S
1
=
H
(
X
)
+
H
(
Y
)
+
H
(
Z
)
and
S
2
=
H
(
X
,
Y
)
+
H
(
Y
,
Z
)
+
H
(
Z
,
X
)
. By the
definition
of
mutual information
,
I
(
X
;
Y
)
+
I
(
Y
;
Z
)
+
I
(
Z
;
X
)
=
2
S
1
−
S
2
,
(1)
so the required right-hand side is
(
2
S
2
−
S
1
)
/3
.
Apply
entropy submodularity
to the
pairs
(
X
,
Y
)
,
(
X
,
Z
)
and then cyclically permute the variables:
H
(
X
,
Y
)
+
H
(
X
,
Z
)
H
(
Y
,
Z
)
+
H
(
Y
,
X
)
H
(
Z
,
X
)
+
H
(
Z
,
Y
)
≥
H
(
X
)
+
H
(
X
,
Y
,
Z
)
,
≥
H
(
Y
)
+
H
(
X
,
Y
,
Z
)
,
≥
H
(
Z
)
+
H
(
X
,
Y
,
Z
)
.
(2)
Adding gives
2
S
2
≥
S
1
+
3
H
(
X
,
Y
,
Z
)
, which is exactly
H
(
X
,
Y
,
Z
)
≤
2
1
S
2
−
6
1
(
2
S
1
−
S
2
)
.
(3)
Solved by
gpt-5
.
6
-sol high.
Ancestors
(10)
1
Paper 164
iii
2025
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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