 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 207

With administrative censoring at $c$, one patient is observed to experience $A$ with probability
$$
\int_0^c\theta e^{-(\theta+\phi)t}\,dt
=\frac{\theta}{\theta+\phi}
\left(1-e^{-(\theta+\phi)c}\right),
$$
so
$$
\mathbb E[v_+]
=n\frac{\theta}{\theta+\phi}
\left(1-e^{-(\theta+\phi)c}\right).
$$
The observed time is $\min(T_A,T_B,c)$, whose mean follows from the <tail-sum formula for expectation>:
$$
\mathbb E[\min(T_A,T_B,c)]
=\int_0^ce^{-(\theta+\phi)t}\,dt
=\frac{1-e^{-(\theta+\phi)c}}{\theta+\phi}.
$$
Hence
$$
\mathbb E[x_+]
=n\frac{1-e^{-(\theta+\phi)c}}{\theta+\phi},
\qquad
\frac{\mathbb E[v_+]}{\mathbb E[x_+]}=\theta.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 6 / b / iv / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 6 / b / iv / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 6 / b / iv / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)