Consider the following transition probability graph:
This figure depicts a Markov chain with three possible states. The possible states are S_1, S_2, and S_3, which are depicted as a row of circles on the middle of the diagram and placed from left to right in this order. At the upper part of the diagram, there are self-loops within S_1, S_2, and S_3, which are circular arrows with both the tail and the head at the same state. In this figure, transitions are depicted by arrows from a circle (representing the source) to a circle (representing the destination). The source and the destination may be the same, and transition arrows may be labelled with a number or a variable indicating the probability that if we start at the source, we will be at the given destination at the next step. The probabilities for each of the self-loops are 0.6 for S_1, 0.7 for S_2, and 0.4 for S_3. Between the circles representing the states are transitions between S_1 and S_2, as well as transitions between S_2 and S_3. The transition probability from S_1 to S_2 is 0.1, and that from S_2 to S_1 is 0.2. The transition probability from S_2 to S_3 is 0.1, and that from S_3 to S_2 is 0.3. Below the row where the states are presented, the transitions between S_1 and S_3 are shown. The transition probe ability from S_1 to S_3 and from S_3 to S_1 are both 0.3.
Calculate the three-step transition probability
r11(3) by using the recursion formula rij(n)=3∑k=1 rik(n−1)pkj.
0.419
Oh boy, looks like you're taking me on a journey through Markov chains and transition probabilities! Alright, let's see if I can help with the calculations.
We want to find the three-step transition probability r11(3) using the recursion formula. According to the formula rij(n) = ∑k=1 rik(n-1)pkj, we need to calculate the probability of transitioning from state S1 to S1 in three steps.
Let's break it down step by step:
Step 1: r11(1)
Since we're starting with one step, we take the direct transition probability from S1 to S1, which is 0.6. So r11(1) = 0.6.
Step 2: r11(2)
Now we move to the second step. We need to consider all possible paths that lead us from S1 to S1 in two steps. There are two options: S1 -> S1 (with probability 0.6) and S1 -> S2 -> S1 (with probability 0.1 * 0.2).
Using the formula, we calculate:
r11(2) = r11(1) * p11 + r12(1) * p21
= 0.6 * 0.6 + 0 * 0.2
= 0.36
Step 3: r11(3)
Finally, we reach the third step. Similarly to step 2, we consider all possible paths that lead us from S1 to S1 in three steps.
There are three options:
1. S1 -> S1 -> S1 (with probability 0.6 * 0.6)
2. S1 -> S2 -> S2 -> S1 (with probability 0.1 * 0.7 * 0.2)
3. S1 -> S3 -> S1 (with probability 0.3 * 0.3)
Plugging these values into the formula:
r11(3) = r11(2) * p11 + r12(2) * p21 + r13(2) * p31
= 0.36 * 0.6 + 0 * 0.2 + 0 * 0.3
= 0.216
And there you have it! The three-step transition probability r11(3) is 0.216. I hope my clownish explanation didn't make things too confusing.
To calculate the three-step transition probability r11(3) using the recursion formula rij(n) = ∑ k=1^3 rik(n-1)pkj, we need to substitute the given values into the formula.
In this case, we are interested in calculating r11(3), which represents the probability of transitioning from state S1 to state S1 in three steps.
Using the recursion formula, we can write the equation as:
r11(3) = r11(2)p11 + r12(2)p21 + r13(2)p31
To calculate the above probabilities, we need to calculate r11(2), r12(2), and r13(2) first.
r11(2) = r11(1)p11 + r12(1)p21 + r13(1)p31
= (0.6)(0.6) + (0.1)(0.2) + (0.3)(0.3)
= 0.36 + 0.02 + 0.09
= 0.47
r12(2) = r11(1)p12 + r12(1)p22 + r13(1)p32
= (0.6)(0.1) + (0.1)(0.7) + (0.3)(0.1)
= 0.06 + 0.07 + 0.03
= 0.16
r13(2) = r11(1)p13 + r12(1)p23 + r13(1)p33
= (0.6)(0.3) + (0.1)(0) + (0.3)(0.4)
= 0.18 + 0 + 0.12
= 0.30
Now we can substitute these values back into the original equation to calculate r11(3):
r11(3) = r11(2)p11 + r12(2)p21 + r13(2)p31
= (0.47)(0.6) + (0.16)(0.2) + (0.30)(0.3)
= 0.282 + 0.032 + 0.09
= 0.404
Therefore, the three-step transition probability r11(3) is 0.404.
To calculate the three-step transition probability r11(3) using the recursion formula, we will use the given equation:
rij(n) = ∑k=1^3 rik(n-1) * pkj
Step 1: Initialize the transition matrix R(0) with the initial probabilities of each state:
R(0) = [r11(0), r12(0), r13(0)]
[r21(0), r22(0), r23(0)]
[r31(0), r32(0), r33(0)]
In this case, since the transition probabilities are provided, we have:
R(0) = [1, 0, 0]
[0, 1, 0]
[0, 0, 1]
Step 2: Calculate R(1) using the recursion formula:
R(1) = R(0) * P
where P is the transition probability matrix:
P = [0.6, 0.1, 0.3]
[0.2, 0.7, 0.1]
[0.3, 0.1, 0.4]
So,
R(1) = R(0) * P
= [1, 0, 0] * [0.6, 0.1, 0.3]
[0.2, 0.7, 0.1]
[0.3, 0.1, 0.4]
= [0.6, 0.1, 0.3]
Step 3: Calculate R(2) using the recursion formula:
R(2) = R(1) * P
R(2) = [0.6, 0.1, 0.3] * [0.6, 0.1, 0.3]
[0.2, 0.7, 0.1]
[0.3, 0.1, 0.4]
= [0.37, 0.24, 0.39]
Step 4: Calculate R(3) using the recursion formula:
R(3) = R(2) * P
R(3) = [0.37, 0.24, 0.39] * [0.6, 0.1, 0.3]
[0.2, 0.7, 0.1]
[0.3, 0.1, 0.4]
= [0.331, 0.208, 0.461]
Therefore, the three-step transition probability r11(3) is approximately 0.331