Consider a Markov chain X0,X1,X2,É described by the transition probability graph shown below. The chain starts at state 1; that is, X0=1.
1recurs p=.75
1to 2 p= .25
2to 1 p = .375
2 recurs p=.25
2 to 3 p = .375
3 to 2 p = .25
3 recurs p = .75
Find the probability that X2=3.
P(X2=3)= 5/8 - incorrect
Find the probability that the process is in state 3 immediately after the second change of state. (A Òchange of state" is a transition that is not a self-transition.)
1 - incorrect
Find (approximately) P(X1000=2∣X1000=X1001).
P(X1000=2∣X1000=X1001)Å 1/4 - incorrect
Let T be the first time that the state is equal to 3.
E[T]= 5 - incorrect
Suppose for this part of the problem that the process starts instead at state 2, i.e., X0=2. Let S be the first time by which both states 1 and 3 have been visited.
E[S]= 6 - incorrect
1 = 3/32
2 = 1/2
3 =
4 =
5 =
1. 3/32
2. 0.5
3. 1/10
4. 32/3
5. 12
Thank you all the above answers are correct
To solve these probability questions related to the given Markov chain, we can use the concept of transition probabilities and the properties of Markov chains. Let's go step by step:
1. Probability that X2=3:
To find the probability that X2=3, we need to consider all possible paths leading to state 3 at time 2. There are two such paths: 1 -> 2 -> 3 and 1 -> 1 -> 3.
Using the given transition probabilities, we have:
P(X2=3) = P(X1=2, X2=3) + P(X1=1, X2=3)
= P(X0=1, X1=2) * P(X1=2, X2=3) + P(X0=1, X1=1) * P(X1=1, X2=3)
= 0.25 * 0.375 + 0.75 * 0.375
= 0.09375
Therefore, the correct probability is P(X2=3) = 0.09375.
2. Probability that the process is in state 3 immediately after the second change of state:
A change of state is a transition that is not a self-transition. In this case, the only possible change of state is from 1 to 2.
Using the given transition probabilities, we have:
P(State 3 immediately after the second change of state) = P(X1=2, X2=3)
= P(X0=1, X1=2) * P(X1=2, X2=3)
= 0.25 * 0.375
= 0.09375
Therefore, the correct probability is 0.09375.
3. Probability P(X1000=2|X1000=X1001):
To find this probability, we need to consider the equation P(X1000=2, X1000=X1001) / P(X1000=X1001).
Using the given transition probabilities, we have:
P(X1000=2, X1000=X1001) = P(X999=2, X1000=2, X1000=X1001)
= P(X998=2, X999=2) * P(X999=2, X1000=2, X1000=X1001)
= P(X998=2, X999=2) * P(X999=2, X1000=X1001| X999=2, X1000=2) * P(X1000=2 | X999=2)
P(X1000=X1001) = P(X999=X1001, X1000=X1001)
= P(X998=X1001, X999=X1001) * P(X999=X1001, X1000=X1001 | X998=X1001, X999=X1001)
Therefore, P(X1000=2|X1000=X1001) = P(X1000=2, X1000=X1001) / P(X1000=X1001)
= P(X998=2, X999=2) * P(X999=2, X1000=X1001| X999=2, X1000=2) * P(X1000=2 | X999=2) / (P(X998=X1001, X999=X1001) * P(X999=X1001, X1000=X1001 | X998=X1001, X999=X1001))
To approximate the value of this probability, we need additional information or assumptions about the initial distribution of the Markov chain.
4. Expected time until the state is equal to 3 (T):
To find this, we need to consider the expected value of the random variable T, which represents the time until the state is equal to 3.
Using the given transition probabilities, we can define the recursive relationship E[T] = (1 - p_33) + p_33 * (E[T] + 1), where p_33 is the transition probability from state 3 to 3.
Simplifying the equation, we get E[T] = (1 - p_33) / (1 - p_33)
Since the value of p_33 is not given in the question, we cannot determine the exact value of E[T] without that information.
5. Expected time until both states 1 and 3 have been visited (S):
To find this, we need to consider the expected value of the random variable S, which represents the time until both states 1 and 3 have been visited.
Using the given transition probabilities, we can define the recursive relationship E[S] = 1 + (1 - p_31) * E[S] + (1 - p_32) * E[S], where p_31 is the transition probability from state 3 to 1, and p_32 is the transition probability from state 3 to 2.
Simplifying the equation, we get E[S] = 1 / (1 - p_31 - p_32)
Since the values of p_31 and p_32 are not given in the question, we cannot determine the exact value of E[S] without that information.