Linear Algebra-Markov's Processes

[charlies1902]

Is anyone on here familiar with Markov Processes?

We're learning about this in linear algebra and I am unsure whether I'm doing this problem right. I attached the problem.

for part a) my matrix is a 2x2 T=
.7 .2
.3 .8

part b)
for 2 years I multiply T^2 * V
where V=[.35 .65]^t
This yields [.3875 .6125]^t
So the # of commuters using mass transit is .3875?


Did I do this problem right? It seems weird to be talking about the "# of commuters" as anything but a whole #.

 

[Ray Vickson]

Is anyone on here familiar with Markov Processes?

We're learning about this in linear algebra and I am unsure whether I'm doing this problem right. I attached the problem.

for part a) my matrix is a 2x2 T=
.7 .2
.3 .8

part b)
for 2 years I multiply T^2 * V
where V=[.35 .65]^t
This yields [.3875 .6125]^t
So the # of commuters using mass transit is .3875?

Did I do this problem right? It seems weird to be talking about the "# of commuters" as anything but a whole #.

Why do you represent the matrix with columns summing to 1? Almost all treatments of Markov processes (at least in probability applications) have the rows summing to 1, so they would use the matrix
$$\pmatrix{.7 & .3\\.2 & .8}$$

Anyway, the elements of V represent "fractions", so 0.3875 means that 38.75% of commuters use mass transit.

RGV

 

[charlies1902]

Really? We were instructed in class that the sum of each column should add up to one. THe solutions for some of the problems from the book add up to 1 as well. Maybe it is different for lienar algebra?

 

[Ray Vickson]

Really? We were instructed in class that the sum of each column should add up to one. THe solutions for some of the problems from the book add up to 1 as well. Maybe it is different for lienar algebra?

Yes, really. See, eg., http://www.ams.org/bookstore/pspdf/mbk-58-prev.pdf or
http://www.classes.cs.uchicago.edu/archive/2005/fall/27100-1/Markov.pdf or
http://people.brandeis.edu/~igusa/Math56aS08/Math56a_S08_notes011.pdf or
http://www.aw-bc.com/greenwell/markov.pdf .

Some other web pages use the column-sums = 1 convention, but the majority use the row-sums=1. I think that the column-sum = 1 convention is more common in Asia than in North America or Europe.

Basically, though, you should use whatever convention your course instructor uses.

RGV