Finding the steady state vector and probablity confused, matrices.

[mr_coffee]

Hello everyone, confused. the directions to this problem are the following:
Find the steay-steat vector, and assuming the chain starts at 1, find the probablity that it is in state 2, after 3 transitions.
well i got the problem and i got the S0 to S3, because it said after 3 transitions, is that what they wanted ,or did they want the lorn term steady state vector? also how do i find the probabliy?> Thanks.
Picture is here:
http://show.imagehosting.us/show/758415/0/nouser_758/T0_-1_758415.jpg

 

[Nate810]

The steady-state vector is the long-term probability distribution of the Markov chain, which tells you the probability that the chain will be in each state after a large number of transitions. To find the steady-state vector, you need to solve the system of linear equations: pi * P = piwhere pi is the steady-state vector and P is the transition matrix. In your case, the transition matrix is: P = [0.6 0.4; 0.5 0.5]and the steady-state vector should satisfy the equation: [p1; p2] * [0.6 0.4; 0.5 0.5] = [p1; p2]Solve this equation to find the steady-state vector.To find the probability that the chain is in state 2 after 3 transitions, you can use the transition matrix. The probability that the chain is in state 2 after 3 transitions is equal to the element in the second row and third column of P^3 (where P^3 is the cube of the transition matrix). In your case, P^3 = [0.51 0.49; 0.51 0.49], so the probability that the chain is in state 2 after 3 transitions is 0.49.

 

[quicknote]

Hello, it appears that you are working on a problem related to Markov chains and their steady state vectors. The steady state vector is the long-term probability distribution of a Markov chain, where each element represents the probability of being in a particular state after an infinite number of transitions. In order to find the steady state vector, you would need to set up and solve a system of equations using the transition probabilities between each state.

In this specific problem, it seems like the instructions are asking for the steady state vector after 3 transitions, which would be the probability distribution after 3 steps. This can be found by multiplying the initial state vector (starting at 1) by the transition matrix 3 times.

To find the probability of being in state 2 after 3 transitions, you would need to look at the second element in the steady state vector. This represents the probability of being in state 2 after an infinite number of transitions, but since we are only looking at 3 transitions, you would need to take the second element in the steady state vector and divide it by the sum of all the elements in the vector to get the probability.

I hope this helps clarify the problem for you. Remember to always double check the instructions and assumptions before starting a problem. Good luck!