All right in this video I'm going to try to do two problems about steady-state
distribution vectors. I'm going to do one example where the transition matrix
for the Markov chain is a 2x2 which is this one and then I'll try to do one
where the transition matrix is 3x3. This problem says, find the steady-state
distribution vector for the regular Markov chain whose transition matrix is
the one given. When you're given the transition matrix, the steady-state
vector equation that you want to use is going to be TL equals L where T is a
transition matrix and then L will be the vector that you're looking for. So in
this case because this is a 2x2 matrix let's just start off by saying L
is going to be x and y and then we're going to solve for what x and y are in
the problem. We have T which is 0.7 0.2 0.3 0.8 times
x y equals x y. The other condition that we know has to hold and that we need to
find a unique solution to x and y is that x and y has to be equal to 1.
The probabilities always have to be equal to 1 in the columns and it's the
same thing for the steady-state distribution vector. Okay so that's going
to produce three equations. We have x and y equals 1. Then we have 0.7 x plus
0.2 y equals x, just multiplying. Then we have 0.3 x plus 0.8 y equals y. This
gives us three linear equations that we can use to find unique solutions for x
and y. The next thing we want to do is move x to the left side and move y to
the left side so that we can have 0 here and 0 here, then we can rewrite the
coefficients in a matrix, do row reduction, and that will give us the
solution. By that I just mean we have this again. Then we'll have -0.3
x plus 0.2 y equals 0 and we have 0.3 x minus 0.2 y equals 0. Let's rewrite that
as a matrix. So we have 1 1 equals 1. Then we have
-0.3 0.2 equals 0 and we have 0.3 -0.2 equals 0. If you
have to solve this by hand, you would use the Gauss-Jordan elimination method or
basically just do row reduction in your calculator. I think I'll do a video on
how to simplify this in another video but for now just plug it into your
calculator. To put this in your calculator, you just do second matrix
which is the inverse symbol, you edit the matrix that you want, then you go to math.
You want "rref". Put in the matrix and then that'll give you the output. The
outcome that I get is 1 0 .4, 0 1 .6, and 0 0
0. You can interpret that as meaning x equals 0.4, y equals
0.6. Your answer written as the limiting vector or steady-state vector
is 0.4 0.6. Sometimes instead of vectors they're called steady-state matrices in
which case you just write it 0.4 0.6 and you just repeat the column
again as many columns as is in your transition matrix but it's the same
thing. Okay so then I'll do the 3x3 matrix example next. Okay here's the
second example. This one's a 3x3 matrix but the idea is still the
same. Again we're going to want to do TL equals L and then find L. The
difference is this time L will have three variables x y z but the concepts are
still the same as the other problem. We have .6 .4 .3, .3 .3 .3,
.1 .3 .4 times x y z equals x y z. That's going to be 0.6 x
plus 0.4 y plus 0.3 z equals x. 0.3 x plus 0.3 y plus 0.3 z equals y. 0.1 x
plus 0.3 y plus 0.4 z equals z and we also have x plus y plus z equals 1.
Rewriting that, we get -0.4 x plus 0.4 y plus 0.3 z equals 0. You
get 0.3 x minus 0.7 y plus 0.3 z equals 0. We get
0.1 x plus 0.3 y minus 0.6 z equals 0 and we have x plus y plus z equals 1.
Rewriting that as a matrix, we get -0.4 0.4 0.3 equals 0. 0.3
-0.7 0.3 equals 0. 0.1 0.3 -0.6 equals 0 and 1 1 1 equals 1.
Let's put that in the calculator and do row reduction. The values in the matrix
table that it will give you are in decimals if you hit math and then 1 that
will change the answer to a fraction and if you hit enter again then you'll get x
equals 33 over 70, y equals 3 over 10, and z equals 8 over 35. Your steady state
matrix or vector is 33/70 3/10 8/35 and we're done with this one as well.