Predicted distribution in x generations?

[Rijad Hadzic]

Homework Statement

Consider the stochastic matrix P giving the probabilities of non-college educated households having at least one college educated child.
<br /> \begin{pmatrix}<br /> .9 &amp; .25\\<br /> .1 &amp; .75 \\<br /> \end{pmatrix}<br />

If there are currently 300,000 college educated households, and 750000 noncollege educated households, in DFW, what is the predicted distribution in 10 generations?

Homework Equations

The Attempt at a Solution

so I have
P =
<br /> \begin{pmatrix}<br /> .9 &amp; .25\\<br /> .1 &amp; .75 \\<br /> \end{pmatrix}<br />

and
x_0
<br /> \begin{pmatrix}<br /> 300\\<br /> 750\\<br /> \end{pmatrix}<br />

So the predicted distribution in 10 generations will be P^9 * x_0, right? Not P^10 * x_0

Sorry I know its probably a simple question but I'm just trying to make sense of this.

 

[Ray Vickson]

Homework Statement

Consider the stochastic matrix P giving the probabilities of non-college educated households having at least one college educated child.
<br /> \begin{pmatrix}<br /> .9 &amp; .25\\<br /> .1 &amp; .75 \\<br /> \end{pmatrix}<br />

If there are currently 300,000 college educated households, and 750000 noncollege educated households, in DFW, what is the predicted distribution in 10 generations?

Homework Equations

The Attempt at a Solution

so I have
P =
<br /> \begin{pmatrix}<br /> .9 &amp; .25\\<br /> .1 &amp; .75 \\<br /> \end{pmatrix}<br />

and
x_0
<br /> \begin{pmatrix}<br /> 300\\<br /> 750\\<br /> \end{pmatrix}<br />

So the predicted distribution in 10 generations will be P^9 * x_0, right? Not P^10 * x_0

Sorry I know its probably a simple question but I'm just trying to make sense of this.

As I interpret it, the distribution In one generation is $x_1 = P x_0$, etc. However, you should use whatever convention your instructor/textbook uses.

Perhaps this is a bit like the difference between building descriptions in North America and Britain: in North America the first floor is on the ground level, but in Britain it is one floor above the ground.

 

[Rijad Hadzic]

As I interpret it, the distribution In one generation is $x_1 = P x_0$, etc. However, you should use whatever convention your instructor/textbook uses.

Perhaps this is a bit like the difference between building descriptions in North America and Britain: in North America the first floor is on the ground level, but in Britain it is one floor above the ground.

Alright. I think your interpretation is correct. I remember my instructor mentioning something about this but my memory is not perfect so I wasnt sure, but that makes sense to me, Since you start with an inital population, and that times P^1 = your first generation...