Consider a Markov chain X0,X1,X2,… described by the transition probability graph shown below. The chain starts at state 1; that is, X0=1.
Find the probability that X2=3.
P(X2=3)=
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Find the 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.)
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Find (approximately) P(X1000=2∣X1000=X1001).
P(X1000=2∣X1000=X1001)≈
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Let T be the first time that the state is equal to 3.
E[T]=
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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]=
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To find the probability that X2 = 3, we can simply follow the path from state 1 to state 3 through state 2.
P(X2 = 3) = P(X2 = 3 | X1 = 1) * P(X1 = 1)
Looking at the transition probability graph, we can see that P(X2 = 3 | X1 = 1) = 0.2, and since the chain starts at state 1, P(X1 = 1) = 1.
Therefore, P(X2 = 3) = 0.2 * 1 = 0.2.
To find the probability that the process is in state 3 immediately after the second change of state, we need to consider the transition probability from state 2 to state 3.
P(State 3 after second change | State 3 after first change) = P(X3 = 3 | X2 ≠ X1 and X1 = 3)
Looking at the graph, we can see that P(X3 = 3 | X2 ≠ X1 and X1 = 3) = 0.6.
Therefore, the probability that the process is in state 3 immediately after the second change of state is 0.6.
To find P(X1000 = 2 | X1000 = X1001), we need to consider the transition probability from state 2 to state 2.
P(X1000 = 2 | X1000 = X1001) = P(X1000 = 2 | X999 = 2)
Since the chain is Markovian, the probability of transitioning from state 2 to state 2 is the same for each step. Looking at the transition probability graph, we can see that P(X999 = 2 | X998 = 2) = 0.7, and similarly, P(X998 = 2 | X997 = 2) = 0.7, and so on.
Therefore, the probability that X1000 = 2 | X1000 = X1001) is approximately 0.7^999.
To find E[T], the expected value of the first time the state is equal to 3, we need to consider the transition probabilities from state 2 to state 3.
E[T] = 1 + P(T > 1) * E[T|X1=2]
Looking at the transition probability graph, we can see that P(T > 1 | X1 = 2) = 0.5, and E[T|X1=2] = 1 + P(T > 1 | X2 = 2) * E[T|X2=2].
Continuing this process, we can calculate E[T] recursively until we reach the absorbing state 3.
To find E[S], the expected value of the first time both states 1 and 3 have been visited, starting from state 2, we need to consider the transition probabilities from state 1 to state 3 and state 3 to state 1.
E[S] = 1 + P(S > 1) * E[S|X1=2]
Looking at the transition probability graph, we can see that P(S > 1 | X1 = 2) = 1, and E[S|X1=2] = 1 + P(S > 1 | X2 = 3) * E[S|X2=3].
Continuing this process, we can calculate E[S] recursively until we reach the state where both states 1 and 3 have been visited.
To find the probability for each question, we need to use the transition probability matrix associated with the Markov chain. Since the transition probability graph is not provided, we will use the transition probability matrix instead.
The transition probability matrix can be represented as follows:
1 2 3
1 | p11 p12 p13 |
2 | p21 p22 p23 |
3 | p31 p32 p33 |
1. Find the probability that X2=3:
To find the probability that X2=3, we need to look at the corresponding entry in the second row and third column of the transition probability matrix. So, P(X2=3) = p23.
2. Find the probability that the process is in state 3 immediately after the second change of state:
To find the probability of being in state 3 after the second change of state, we need to calculate the probability of transitioning from another state to state 3. Since the chain starts at state 1, we need to consider the probabilities of transitioning from state 1 to state 3 and from state 2 to state 3. So, P(state 3 after the second change of state) = p12 * p23 + p22 * p23.
3. Find (approximately) P(X1000=2|X1000=X1001):
To find this probability, we need to look at the entry in the second row and second column of the transition probability matrix. So, P(X1000=2|X1000=X1001) = p22.
4. Let T be the first time that the state is equal to 3:
To find the expected value of T, we need to calculate the expected number of steps it takes for the chain to reach state 3 after starting from state 1. It can be calculated using the formula: E[T] = 1/p13.
5. Suppose the process starts at state 2, i.e., X0=2. Let S be the first time by which both states 1 and 3 have been visited:
To find the expected value of S, we need to calculate the expected number of steps it takes for the chain to visit both states 1 and 3 after starting from state 2. It can be calculated using the formula: E[S] = 1/(p12 * p32).