Final Examination
(只有二題)※ 請在 1 月 16 日晚上 12 點前 e-mail 給我期末考答案卷。
※ 期中考該繳交的資料也請在 1 月 16 日晚上 12 點前給我。
※ 期末考答案卷內使用的重要定理也請一併列出證明，答案卷得分的高低乃根據解題詳細狀況而定。
Definition we say that a row vector v=( ,v v v1 2, 2,..., )vt is a distribution vector if v v v1, 2, 2,...,vt ≥0 and v1+ + + + =v2 v2 ... vt 1
Problem 1. Let G be a non-bipartite connected graph with n vertices and m edges, and {X0,X X1, 2,...} be a random walk on G. Show that the distribution of Xk tends to a stationary distribution of the transition matrix of the random walk.
Problem 2. Let P =[pij] be a transition matrix of a Markov chain which has state space S. Suppose that
ij 0
p > for each i, j in state space S. Show that, for any distribution vector
