course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Markov Chains |
II |
38 |
|
A3.1 B3.1 |
2002 |
(i) Consider the continuous-time Markov chain
Compute the probability, starting from state 3 , that
Deduce that
$$\lim {t \rightarrow \infty} \mathbb{P}\left(X{t}=2 \mid X_{0}=3\right)=\frac{4}{15}$$
[Justification of standard arguments is not expected.]
(ii) A colony of cells contains immature and mature cells. Each immature cell, after an exponential time of parameter 2, becomes a mature cell. Each mature cell, after an exponential time of parameter 3, divides into two immature cells. Suppose we begin with one immature cell and let