Stat3021 Assignment 1

1.假设Xn是独立的并且等价分布为P（X1 = 1）= p，
P（X1 = 0）= r和P（X1 = -1）= q。其中p，r，q> 0且p + r + q = 1。

n
i = 1 Xi
，n = 1，2，。 。 ..
（a）证明{S1，S2，。 。 。}是状态空间为S =的不可约马尔可夫链
{0，±1，±2，…}并记下其转移矩阵。
（b）链条是非周期性的吗？
（c）查找以下表达式：

ii。 P（S4 = 1 | S1 = 1）。
iii。 P（S10 = 1 | S7 = 0）。
iv。 ESn和var（Sn）。
2.假定过渡图如下：

（a）求出状态3的时间段以及从状态3开始的

（b）假设X0 = 3，求出链被R1吸收的概率。
（c）假设X0 = 3，找到直到链的预期时间（步数）

E [T | X0 = 3]。

3. An apparatus consists of two containers C1 and C2, and four balls, in one or the
other of the two containers. At regular times the following procedure is carried out:
with probability θ a ball is chosen uniformly at random (from all four balls) and
is removed from its container and placed in the other one; with probability 1 − θ
things are left as they were. Let Xn denote the number of balls in container C1 at
time n.
(a) Explain why Xn forms a Markov chain with stationary transition probabilities.
(b) Complete the following transition matrix P
P =


1 − θ θ ∗ ∗ ∗
θ/4 1 − θ 3θ/4 ∗ ∗
0 θ/2 ∗ ∗ ∗
∗ ∗ ∗ 1 − θ θ/4
∗ ∗ ∗ ∗ 1 − θ


(c) Classify the states, taking care to distinguish the cases θ = 0, 0 < θ < 1 and
θ = 1.
Draw a transition diagram to aid your explanation.
(d) Suppose 0 < θ < 1. Calculate the stationary distribution π = (π0, · · · , π4).
Explain briefly why the distribution is the same for all θ ∈ (0, 1).
(e) Suppose 0 < θ < 1. What is the limit of P
n
, where P is the transition matrix
given in part (b)? Explain.
(f) Let mj be the expected time to reach state 4, given X0 = j, for j = 0, 1, 2, 3.
Calculate mj when θ = 0