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.
