Xn is a Markov Chain with state-space
E = {0, 1, 2}, and transition matrix
0.4 0.2 0.4
P = 0.6 0.3 0.1
0.5 0.3 0.2
And initial probability vector a = [0.2, 0.3,
0.5]
For the Markov Chain with state-space, initial vector, and
transition matrix discuss how we would calculate the follow;
explain in words how to calculate the question below.
a) P(X1 = 0, X2
= 0, X3 = 1, X4 =
2|X0 = 2)
b) P(X2 =...
Xn is a discrete-time Markov chain with state-space {1,2,3},
transition matrix, P =
.2
.1
.7
.3
.3
.4
.6
.3
.1
a) find E[X1|X0=2] =
b) The P(X9=1|X7=3) =
C) The P(X2=2) =
Q5. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1,
2, 3}, and transition probability matrix (pij ) given by 2 3
1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2 Determine all
recurrent states. 1 2 Q6. Let {Xn|n ≥ 0} is a Markov chain with
state space S = {0, 1, 2} and transition...
Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2,
3}, and transition probability matrix (pij ) given by 2 3 1 3
0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2 Determine all
recurrent states. Q3. Let {Xn|n ≥ 0} is a Markov chain with state
space S = {0, 1, 2} and transition probability matrix (pij...
Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1,
2, 3} and transition probability matrix (pij ). Let τi = min{n ≥ 1
: Xn = i}, i = 0, 1, 2, 3. Define Bij = {Xτj = i}. Is Bij ∈ σ(X0, ·
· · , Xτj ) ? Q2. Let {Xn|n ≥ 0} is a Markov chain with state space
S = {0, 1, 2, 3}, X0 = 0, and transition...
Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S. For i ∈
S, define τi = min{n ≥ 0|Xn = i}. Show that τi is a stopping time
for each i. Q2. Let τi as in Q1. but for any discrete time
stochastic process. Is τi a stopping time? Q3. Let {Xn|n ≥ 0} be a
Markov chain and i is a state. Define the random time τ = min{n ≥
0|Xn+1 = i}. If τ...
Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, ·
· · } and transition matrix (pij ) given by pij = 1 2 if j = i − 1
1 2 if j = i + 1, i ≥ 1, and p00 = p01 = 1 2 . Find P{X0 ≤ X1 ≤ · ·
· ≤ Xn|X0 = i}, i ≥ 0
. Q2. Consider the Markov chain given in Q1. Find P{X1,...
Let Xn be a Markov chain with states 0,1,...,9 and transition
probabilities P0,0 = P0,1 = P9,8 = P9,9 = 1/2 an Pi,i = Pi,i+1 =
Pi,i−1 = 1/3 for all 1 ≤ i ≤ 8.
(a) Draw the transition diagram.
(b) What is the probability that X1,X2,X3,X4 are all smaller
than 3 given that X0 = 1?
Hint: Create a simpler Markov chain with 4 states.
Let Xn be the Markov chain with states S = {1, 2, 3, 4} and
transition matrix.
1/3
2/3
0
0
2/3
0
1/3
0
1/3
1/3
0
1/3
0
1/3
2/3
0
a.) Let X0 = 3 and let T3 be the first time that the Markov
chain returns 3, compute P(T3 = 2 given X0=3). Please show all work
and all steps.
b.) Find the stationary distribution π. Please show all work and
all steps.