Astate vector x for a four-state markov chain is such that the system is four times as likely to be in state 2 as in 4, is not in state 3, and is in state 1 with probability 0.2. find the state vector

All the components in the state vector need to sum to 1. You're given that component corresponding to state 1 is 0.2, and that the component for state 3 is 0.

That leaves states 2 and 4, for which you're told that the component for state 2 is four times as large. If is the component for state , then you have



which means . So the state vector is .