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Past exam of the mathematics course of the University of Cambridge
/
2023
/
iii
/
Paper 356
/
3
/
c
/
Solution
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Past exam of the mathematics course of the University of Cambridge
2023
iii
Paper 356
3
c
2026-09-28
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Averaging the slow propensities over the conditional
Poisson distribution
uses
E
[
Y
(
Y
−
1
)
∣
X
=
x
]
=
q
(
x
)
2
. Thus the effective birth and
death
rates
of
X
are
λ
1
(
x
)
=
V
α
1
q
(
x
)
2
=
α
4
2
x
2
α
1
α
3
2
V
3
,
(1)
λ
2
(
x
)
=
V
α
2
x
(
x
−
1
)
.
(2)
Consequently
∂
t
p
0
(
x
,
t
)
=
([
E
x
−
1
−
1
]
λ
1
(
x
)
+
[
E
x
+
1
−
1
]
λ
2
(
x
))
p
0
(
x
,
t
)
.
(3)
Ancestors
(11)
c
3
Paper 356
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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