Mixing
Simple Probability Matrix
Question:
Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your upcoming test. If you fail your previous test, then you have 0.5 chance you will fail your upcoming test. If it continues over a long time, what is the probability that you will pass a test?
I have calculated the eigenvalues and the corresponding eigenvectors of P, but I don't know where to go after that. Any help would be appreciated.
linear-algebra probability matrices random-walk
edited Jan 5 '16 at 23:56
Hussein El Feky
1138
asked Apr 14 '13 at 3:41
swagswag
12
add a comment |
Question:
Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your upcoming test. If you fail your previous test, then you have 0.5 chance you will fail your upcoming test. If it continues over a long time, what is the probability that you will pass a test?
I have calculated the eigenvalues and the corresponding eigenvectors of P, but I don't know where to go after that. Any help would be appreciated.
linear-algebra probability matrices random-walk
edited Jan 5 '16 at 23:56
Hussein El Feky
1138
asked Apr 14 '13 at 3:41
swagswag
12
add a comment |
Question:
Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your upcoming test. If you fail your previous test, then you have 0.5 chance you will fail your upcoming test. If it continues over a long time, what is the probability that you will pass a test?
I have calculated the eigenvalues and the corresponding eigenvectors of P, but I don't know where to go after that. Any help would be appreciated.
linear-algebra probability matrices random-walk
edited Jan 5 '16 at 23:56
Hussein El Feky
1138
asked Apr 14 '13 at 3:41
swagswag
12
Question:
Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your upcoming test. If you fail your previous test, then you have 0.5 chance you will fail your upcoming test. If it continues over a long time, what is the probability that you will pass a test?
I have calculated the eigenvalues and the corresponding eigenvectors of P, but I don't know where to go after that. Any help would be appreciated.
linear-algebra probability matrices random-walk
linear-algebra probability matrices random-walk
edited Jan 5 '16 at 23:56
Hussein El Feky
1138
asked Apr 14 '13 at 3:41
swagswag
12
edited Jan 5 '16 at 23:56
Hussein El Feky
1138
asked Apr 14 '13 at 3:41
swagswag
12
edited Jan 5 '16 at 23:56
Hussein El Feky
1138
edited Jan 5 '16 at 23:56
Hussein El Feky
1138
edited Jan 5 '16 at 23:56
Hussein El Feky
1138
1138
asked Apr 14 '13 at 3:41
swagswag
12
asked Apr 14 '13 at 3:41
swagswag
12
asked Apr 14 '13 at 3:41
swagswag
12
12
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Sounds like you're on the right track. Assuming you've set up the 2x2 matrix that represents the transitions correctly, then you're all set. Recall that the long-run distribution is interpreted as the stationary probability vector, which is the left eigenvector of the transition matrix P associated with eigenvalue 1.
answered Apr 14 '13 at 4:31
daikonradishdaikonradish
1111
I've got the egienvalue 1, and the eigenvector is $vec{w} = [-0.53, -0.848] $, what do you mean when you say "left" eigenvector?
– swag
Apr 14 '13 at 4:50
i mean the vector $pi$ such that $pi P = pi$, so $pi$ is on the left side of $P$, which is your transition matrix. This is not the same as $Ppi = pi$.
– daikonradish
Apr 14 '13 at 5:15
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sounds like you're on the right track. Assuming you've set up the 2x2 matrix that represents the transitions correctly, then you're all set. Recall that the long-run distribution is interpreted as the stationary probability vector, which is the left eigenvector of the transition matrix P associated with eigenvalue 1.
answered Apr 14 '13 at 4:31
daikonradishdaikonradish
1111
I've got the egienvalue 1, and the eigenvector is $vec{w} = [-0.53, -0.848] $, what do you mean when you say "left" eigenvector?
– swag
Apr 14 '13 at 4:50
i mean the vector $pi$ such that $pi P = pi$, so $pi$ is on the left side of $P$, which is your transition matrix. This is not the same as $Ppi = pi$.
– daikonradish
Apr 14 '13 at 5:15
add a comment |
Sounds like you're on the right track. Assuming you've set up the 2x2 matrix that represents the transitions correctly, then you're all set. Recall that the long-run distribution is interpreted as the stationary probability vector, which is the left eigenvector of the transition matrix P associated with eigenvalue 1.
answered Apr 14 '13 at 4:31
daikonradishdaikonradish
1111
I've got the egienvalue 1, and the eigenvector is $vec{w} = [-0.53, -0.848] $, what do you mean when you say "left" eigenvector?
– swag
Apr 14 '13 at 4:50
i mean the vector $pi$ such that $pi P = pi$, so $pi$ is on the left side of $P$, which is your transition matrix. This is not the same as $Ppi = pi$.
– daikonradish
Apr 14 '13 at 5:15
add a comment |
Sounds like you're on the right track. Assuming you've set up the 2x2 matrix that represents the transitions correctly, then you're all set. Recall that the long-run distribution is interpreted as the stationary probability vector, which is the left eigenvector of the transition matrix P associated with eigenvalue 1.
answered Apr 14 '13 at 4:31
daikonradishdaikonradish
1111
Sounds like you're on the right track. Assuming you've set up the 2x2 matrix that represents the transitions correctly, then you're all set. Recall that the long-run distribution is interpreted as the stationary probability vector, which is the left eigenvector of the transition matrix P associated with eigenvalue 1.
answered Apr 14 '13 at 4:31
daikonradishdaikonradish
1111
answered Apr 14 '13 at 4:31
daikonradishdaikonradish
1111
answered Apr 14 '13 at 4:31
daikonradishdaikonradish
1111
answered Apr 14 '13 at 4:31
daikonradishdaikonradish
1111
1111
I've got the egienvalue 1, and the eigenvector is $vec{w} = [-0.53, -0.848] $, what do you mean when you say "left" eigenvector?
– swag
Apr 14 '13 at 4:50
i mean the vector $pi$ such that $pi P = pi$, so $pi$ is on the left side of $P$, which is your transition matrix. This is not the same as $Ppi = pi$.
– daikonradish
Apr 14 '13 at 5:15
add a comment |
I've got the egienvalue 1, and the eigenvector is $vec{w} = [-0.53, -0.848] $, what do you mean when you say "left" eigenvector?
– swag
Apr 14 '13 at 4:50
i mean the vector $pi$ such that $pi P = pi$, so $pi$ is on the left side of $P$, which is your transition matrix. This is not the same as $Ppi = pi$.
– daikonradish
Apr 14 '13 at 5:15
I've got the egienvalue 1, and the eigenvector is $vec{w} = [-0.53, -0.848] $, what do you mean when you say "left" eigenvector?
– swag
Apr 14 '13 at 4:50
I've got the egienvalue 1, and the eigenvector is $vec{w} = [-0.53, -0.848] $, what do you mean when you say "left" eigenvector?
– swag
Apr 14 '13 at 4:50
i mean the vector $pi$ such that $pi P = pi$, so $pi$ is on the left side of $P$, which is your transition matrix. This is not the same as $Ppi = pi$.
– daikonradish
Apr 14 '13 at 5:15
i mean the vector $pi$ such that $pi P = pi$, so $pi$ is on the left side of $P$, which is your transition matrix. This is not the same as $Ppi = pi$.
– daikonradish
Apr 14 '13 at 5:15
add a comment |
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