41. Let i denote the long-run proportion of time a given irreducible Markov chain is in state...

41. Let πi denote the long-run proportion of time a given irreducible Markov chain is in state i.

(a) Explain why πi is also the proportion of transitions that are into state i as well as being the proportion of transitions that are from state i.

(b) πiPij represents the proportion of transitions that satisfy what property?

(c)

i πiPij represent the proportion of transitions that satisfy what property?

(d) Using the preceding explain why

πj =

i

πiPij