課程名稱︰機率導論
課程性質︰數學系大二必修
課程教師︰陳宏
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/06/18
考試時限（分鐘）：60
試題 :
(n)
Notation: P = P(X = j| X = i)
ij n 0
1. (20%) Three white and three black balls are distributed in two urns in such
way that each contains three balls. We say that the system is in state i, i =
0, 1, 2, 3, if the first urn contains i white balls. At each step, we draw
one ball from each urn and place the ball drawn from the first urn into the
second urn, and conversely with the ball from the second urn. Let X denote
n
the state of the system after the n-th step. Explain why {X , n = 0, 1, 2,
n
...} is a Markov chain and calculate its transition probability matrix.
2. Consider a Morkov chain with state space {A, B, C, D, E} and has transition
matrix given by
A B C D E
A 0.1 0.6 0.3 0 0
B 0 0.2 0.3 0.5 0
C 0 0.3 0.2 0.3 0.2
D 0 0 0 0 1
E 0 0 0 1 0
(2) (n)
(a) (10%) Calculate P and P for n ∈ |N.
CA DE
(b) (10%) What are the irreducible classes in this Markov chain?
(c) (10%) Identify the recurrent and transient states.
3. (a) (10%) Let N = #{n ≧ 0, X = y} be the number of times that the process
y n
∞ (n)
is in state y. Show that E[N | X = y] = Σ P .
y 0 n=0 yy
∞ (n)
(b) (10%) Explain why a state y is recurrent if and only if Σ P = ∞.
n=0 yy
(c) (20%) (A Random Walk) Consider a Markov chain whose state space consists
of the integers i = 0, ±1, ±2, ..., and has transition probabilities
given by
P = p, P = 1-p, i = 0, ±1, ±2, ...
i, i+1 i, i-1
where 0 < p < 1. In other words, on each transition the process either
moves one step to the right (with probability p) or one step to the left
(with probability 1-p). Determine whether the state 0 is recurrent or
transient. (Note that it depends on the values of p.)
4. An individual has three umbrellas, some at her office, and some at home. If
she is leaving home in the morning (or leaving work at night) and it is
raining, she will take an umbrella, if one is there. Otherwise, she gets wet.
Assume that independent of the past, it rains on each trip with probability
0.2. To formulate a Markov chain, let X be the number of umbrellas at her
n
current location.
(a) (15%) Find the transition probability for this Markov chain.
(b) (15%) Calculate the limiting fraction of time she gets wet.