MAT 135B Spring 2020
Final Exam
1.第一个问题具有三个独立的部分，但所有三个问题都涉及所示图上的随机游动

b
b
b
1个
2
3
4

1a。 （20分）对图1所示的图形执行离散时间的随机游动。

1b。 （20分）再次在图1的图形上执行离散时间的随机游走。现在

1个

1c。 （20分）在图2中的图形上执行连续时间的随机游动。

1→2 = 1
1→3 = 2
1→4 = 3，
2→1 = 1
2→3 = 2
2→4 = 3，
3→1 = 2
3→2 = 2
3→4 = 3，
4→1 = 3，
4→2 = 3，
4→3 = 3
1.此过程的Q矩阵是什么？
2.考虑转移概率Pt的等式

dt =铂
·Q，P0 =I。

c1 + c2e
−7t + c3e
−9t + c4e
-12吨

2.（20分）假设{Xt}是具有有限状态空间的连续时间马尔可夫过程。让铂表示

dt = Q·Pt = Pt
·Q，P0 =I。（1）
2
1.证明如果概率分布π（行向量）满足π·Pt =π，则π·Q = 0。
2.如果
Q =

−a a

其中a，b> 0，（2）

3. If the two states of the Markov process defined by (2) are labeled 1 and 2, given that the process
starts in state 1, what is the average time the process will remain in state 1 before a transition to
state 2?
3. (20 pts) Let Xt denote the position of a random walker on the integer lattice Z at time t (continuous
time). The nonzero transition rates in the Q-matrix are1
Qx−2,x = p2, Qx−1,x = p1, Qx+1,x = q1, Qx+2,x = q2
where 0 < pi < 1, 0 < qi < 1, and p2 + p1 + q1 + q2 = 1.
1. Find an integral representation of the transition probability2 Py(x;t) with initial condition Py(x; 0) =
δx,y.
2. Show that
E(Xt) = (p2 + p1 − q1 − q2)t + y
where y is the initial position.3
1For simple random walk in continuous time we had q2 = p2 = 0.
2We use the notation for the transition probability y → x used in Notes #11, §2.1.
3Hint: See §2.1.1 of Notes #11.
3