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Past exam of the mathematics course of the University of Cambridge
/
2021
/
iii
/
Paper 201
/
3
/
c
/
Solution
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Past exam of the mathematics course of the University of Cambridge
2021
iii
Paper 201
3
c
2026-09-28
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The
moment-generating function
of one increment is
E
e
θ
ξ
1
=
p
e
θ
+
(
1
−
p
)
e
−
θ
.
(1)
Thus with
ψ
(
θ
)
=
lo
g
(
p
e
θ
+
(
1
−
p
)
e
−
θ
)
,
(2)
independence gives
E
[
Z
n
+
1
∣
F
n
]
=
Z
n
e
−
ψ
(
θ
)
E
e
θ
ξ
n
+
1
=
Z
n
.
(3)
This is the
exponential martingale of a biased simple random walk
.
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(11)
c
3
Paper 201
iii
2021
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
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