SOLVED:A transition probability matrix 𝐏 is said to be doubly stochastic if the sum over each column equals one; that is, ∑i PU=1, for all If such a chain is irreducible and aperiodic and consists of M 1 states 0,1, …, M, show that the limiting probabilities are given by πj=(1)/(M 1), j=0,1, …, M

SOLVED: Consider a DTMC with state space 0, 1, 2, 3 and transition probability matrix:| | 0 | 1 | 2 | 3 ||—|—–|—–|—–|—–|| 0 | 1/3 | 2/3 | 0 | 0 || 1 | 1/2 | 0 | 1/2 | 0 || 2 | 0 | 1/2 | 0 | 1/2 || 3 | 0 | 0 | 2/3 | 1/3 |(a) Find the period of this chain:(b) Are the limiting probabilities and the long-run proportions of the time the DTMC in each state the same for this chain? Why or why not? You do not need to compute those probabilities.

Unused symbols. Show that the capacity of the channel with probability transition matrix Py |x=[ (2)/(3) (1)/(3) 0 (1)/(3) (1)/(3) (1)/(3) 0 (1)/(3) (2)/(3)] is achieved by a distribution that places zero probability on one of input symbols. What is the capacity of this channel? Give an intuitive reason why that letter is not used. | Numerade