Question

Need help with Transition Matrices - Test soon
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.

Answer 1

No idea how to set this out.

Answer 2

The numbers are relatively easy to figure out. But how to set up the transition matrix ...

Answer 3

I wonder if you've seen this video yet: www.youtube.com/watch?v=VeTjArt51Q4

Answer 4

I was thinking
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]

mate what do you think about this?

I just logically filled out the rest. however the 4 column with all 0's im a bit unsure about.

And yes ahaha I listened to that Australian dude last night. It's just this situation is a little more complex. Thanks though.

Answer 5

If I got the matrix right I understand what to do next

T^10 x initial state matrix.

Answer 6

I agree. It appears to be more Complex than what the video explains. I can't promise that what you've done is correct or not though.

Answer 7

It's just setting out the matrix is a bit dodgy. I cant get it right. The text book 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.

Answer 8

What do you think about setting out the worded problem and solving by cancelling out? Could that work?

Answer 9

And then typing in the simplified data into the matrix.

Answer 10

TBH it's really late for me. This question is more complete than what I assumed initially. Furthermore, I don't have the proper tools to engage as much as I want to. Don't worry. There's Someone around Who can help. You might want to try asking your question on Quora.

Answer 11

Complex *

Answer 12

No problem mate. Thanks anyway.

Answer 13

I came up with something similar but a bit different:
c | t | a | d
c [ .875 | 0 | .04 | 0
_
t [.12 | .82| 0 | 0
_
a [0 |.15 | .92| 0
_
d [.005 | .03 | .08 | 0

Basically from yours for row t I have .82 instead of .082 (I think maybe you just typod) ;

For row a column a I have .92 instead of .88 because even though 4 percent of adults are childs being born, adults cannot backtrack into children. so you only subtract adults to death which is .08 .

Give me a sec to type the proof

Answer 14

Yeah I believe thats right, because vertically all the populations sum up to 1.0 except for adults which has a small surplus do to children being born. horizontally you get .915 ; .94 ; 1.07 ; .115 which makes sense because the entire population does not stay as children, and youll have a larger amount of adult population than the inital adult population because the amount of teenagers turning into adults vs adults dieing is higher.