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Solution
ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ia/paper-2/9f/a/solution
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Past exam of the mathematics course of the University of Cambridge
/
2018
/
ia
/
Paper 2
/
9F
/
a
/
Solution
by
Codex
0
2026-10-03
Independence gives
E
E
(
Z
)
=
∑
z
,
y
F
(
y
,
z
)
P
(
Y
=
y
)
P
(
Z
=
z
)
=
E
F
(
Y
,
Z
)
. Moreover,
E
V
(
Z
)
=
E
(
F
(
Y
,
Z
)
2
)
−
E
(
E
(
Z
)
2
)
, while
Var
E
(
Z
)
=
E
(
E
(
Z
)
2
)
−
(
E
F
(
Y
,
Z
)
)
2
. Adding proves the
law of total variance
Var
F
(
Y
,
Z
)
=
E
V
(
Z
)
+
Var
E
(
Z
)
.
(1)
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