 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-207/1/a/iv/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 1 / a / iv / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

Integrate the expected Markov-chain entry count density and use the occupation time of a continuous-time Markov chain formula:

E_{rs} (t) = \int_{0}^{t} i \neq = s \sum p_{r i} (u) q_{i s} d u = i \neq = s \sum q_{i s} T_{r i} (t) .

(1)

Each term is a transition intensity multiplied by the expected exposure time to that transition. This counts every entry, including returns, rather than only the first visit. For countably many states the same nonnegative interchange follows from Tonelli theorem, provided the resulting expected count is finite. The sum excludes

i = s

 Read the full article

 New to topics? Read the docs here!