Transition Matrices - Worded Problem

In summary: So we have 1.04 + 0.82 + 1.00 = 2.86. Dividing by 2.86 makes 0.36, 0.29, 0.35. You can check it by multiplying the matrix by (1,1,1,1). It should give the same as multiplying by the columns.In summary, the conversation discusses a small town with a population that can be grouped into three categories: adults, teenagers, and children. The conversation then delves into statistics on birth and death rates for
  • #1
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A very small country town has a population that can be grouped according to three categories: adults teenagers and children.

Each year statistics show that:

Children are born at the rate of 4% of the adult population 12% of children become teenagers 15% of teenagers become adults 0.5% of children die 3% of teenagers die 8% of adults die

Presuming that the town started with 350 children, 640 teenagers and 2100 adults, find how many there will be of each category after 10 years.

I'm having trouble finding the transition matrix.

So far I've got.
c t a d
c [0.875 0 0.04 0]
t [ 0.12 0.082 0 0]
a [ 0 0.15 0,88 0]
d [ 0.005 0.03 0.08 0]

The textbook somehow gets 0.04 in the 4th row of the 3rd column instead of 0.08 and has a 1 in the 4th row of the 4th column instead of 0.

Not sure how or why.

If I could get the Transition matrix correct I understand that it would simply be

T^10 x initial state but I'm just having trouble setting it out.

Thanks
 

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  • #2
Hello math, welcome to PF :smile: !

$$\begin{bmatrix} c', t', a', d'
\end{bmatrix} =

\begin{bmatrix}0.875 & 0.000 & 0.040 & 0 \\
0.120 & 0.082 & 0.000 & 0 \\
0.000 & 0.150 & 0.880 & 0 \\
0.005 & 0.030 & 0.080 & 0 \\
\end{bmatrix}
\begin{bmatrix} c \\ t \\ a \\ d \\ \end{bmatrix}
$$is the general idea, I hope, where the accents denote a shift by 1 year.

All columns add up to 1, except 2 and 4 in your rendering of the matrix. Whereas the book probably only has column 3 not adding up to 1,
so it looks like an error in the book to me. The 0.04 I can't explain either.
Like your 0.082 seems an error by you...

Funny it's a 4x4 matrix when you only have three categories and the last column is all zeroes.

The book doesn't have that, which indicates to me that once you're dead, you remain dead.
But again, that category isn't counted, so I wouldn't worry.

If still in doubt, let your matrix^10 loose and see if the results are credible !

Isn't it nice to see such a cute society develop: no teenage pregnancies !
 
  • #3
I agree with your .08. For the [d,d] position, dead people stay dead.
How do you get the 0.88? The 0.082 should be 0.82.
 
  • #4
Very good Haru ! Makes me recall the stuff about columns adding up to 1: a category like a can have > 1 because it really the 4% that become parent remain a
 

What are transition matrices?

Transition matrices are mathematical tools used to describe and analyze the movement or transition of objects or individuals from one state to another. They are commonly used in fields such as physics, economics, and biology to model and predict behavior and changes over time.

How do you create a transition matrix?

To create a transition matrix, you first need to identify the different states or categories that an object or individual can transition between. Then, you need to collect data or information on the probability or likelihood of transitioning from one state to another. This information is then organized into a square matrix with the rows and columns representing the different states and the values representing the probabilities of transitioning between them.

What are some real-life applications of transition matrices?

Transition matrices are commonly used in various fields to model and analyze behavior and changes over time. Some examples include predicting the spread of diseases in epidemiology, analyzing customer behavior in marketing, and studying the movement of particles in physics.

What is the difference between a transition matrix and a Markov chain?

A Markov chain is a mathematical model that uses transition matrices to describe the transition of objects or individuals between states. Essentially, a Markov chain is a sequence of transition matrices that represent successive states over time. Therefore, the main difference between the two is that a Markov chain is a dynamic model that takes into account the changes over time, while a transition matrix is a static representation of the probabilities of transitioning between states.

How can transition matrices be used to make predictions?

Transition matrices can be used to make predictions by using the probabilities of transitioning between states to calculate the likelihood of an object or individual being in a certain state at a specific time. This allows for the analysis and prediction of behavior and changes over time, making transition matrices a valuable tool in various fields.

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