## Assignment 5

统计作业代写 i.e. the random times spent in each state before moving and also the probabilities of where the moves go to on leaving each state.

### To be handed in no later than 12 noon on Thursday 9 November

This contributes 10% to the overall mark

For a continuous time Markov process on states *{*1*, *2*, *3*}*, the rate matrix

### for some *λ > *0. The process starts at 2; i.e. *X*(0) = 2. 统计作业代写

1.State the distribution for the number of changes in the process by time *t*. Explain the answer, for which you can use a result we obtained in class for the Poisson process.

2.Describe the process in full; i.e. the random times spent in each state before moving and also the probabilities of where the moves go to on leaving each state.

3.The stationary probability vector *π *satisfies *π P*(*t*) = *π *for all *t*. Show that for such *π *it is that *π G *= 0. Recall that

*P*(*t*) = *e **tG *= *I *+ *tG *+ *t *2*G *2 */*2! + *t *3*G *3 */*3! + *· · · **. 统计作业代写*

4.Find the *π *for the given *G*.

5.If an observer is watching the outcome of the process at an arbitrarty large time point *t*, what is the probability the observer sees the state 3 at that time.

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