(a) Show that the restricting probabilities of the turned around Markov chain are equivalent to for the forward chain by showing that they fulfill the conditions

71)=E7ri Qij (b) Give an instinctive clarification for the consequence of section (a).

Q6

A sum of m white and m debases are dispersed among two urns, with every urn containing m balls. At each stage, a ball is haphazardly chosen from every urn and the two chose balls are traded. Allow X to signify the quantity of repudiates in urn 1 after the nth exchange .

(a) Give the change probabilities of the Markov chain Xn 11 04. (b) Without any calculations, what do you believe are the restricting probabilities of this chain? (c) Find the restricting probabilities and show that the fixed chain is time reversible.

Q7

A gathering of n processors is orchestrated in an arranged rundown. At the point when a task shows up, the main processor in line endeavors it; in the event that it is fruitless, the next attempts it; assuming it also is ineffective, the next attempts it, etc. At the point when the work is effectively handled or after all processors have been fruitless, the work leaves the framework. Now we are permitted to reorder the processors, and a new position offers. Assume that we utilize the one-closer reordering rule, which moves the processor that was effective one nearer to the front of the line by exchanging its situation with the one before it. On the off chance that all processors were fruitless (or assuming the processor in the principal position was effective), the requesting stays as before. Assume that each time processor I endeavors a task at that point, freely of whatever else, it is fruitful with likelihood Pi

(a) Define a proper Markov chain to dissect this model.

(b) Show that this Markov chain is time reversible.

(c) Find the since a long time ago run probabilities.

Q8

A Markov affix is supposed to be a tree cycle in the event that (I) > 0 at whatever point Pei> 0,

00 for each pair of states I and j, I t-j there is a one of a kind grouping of unmistakable states = ich iI inI. in = j to such an extent that

Pik.ik+1 > O. k = 0, 1, n 1 all in all, a Markov chain is a tree cycle if for each pair of unmistakable states I and j there is a one of a kind route for the interaction to go from I to j without returning a state (and this way is the converse of the interesting way from jto t). Contend that an ergodic tree measure is time reversible.