How to create a transition matrix for this problem?

Question

I have a final exam tomorrow and I am stuck on one question on my review sheet. Can someone help explain how to do this problem? Any help would be appreciated.

Here is the problem:

(a) There are two coastal cities that are both beautiful places to live and work. Despite being lovely places to live there is some movement in population between the two cities. Mysteriously every year exactly 4 percent of the inhabitants of cit A decide to move to city B, and 8 percent of city B dwellers decide to move to city A. If people don't move between city A and city B then they just stay in the city they began in. What is the transition matrix that describes this annual migration?

(b) What happens if in addition to people moving around like in part (a), you know that 2 percent of city A dwellers leave the coast for good, and 4 percent of city B dwellers leave the coast for good. What does the transition matrix look like?

I believe the answer to part (a) is:

[0.04, 0.92; 0.96, 0.08]

But I am having trouble with part (b).

Thank you,

Brian

Answer 1

Your answer to a) is wrong. Here's how a transition matrix works: for every $i$ and $j$, $P_{ij}$ (the entry in the $i$th row and $j$th column) is the probability of moving from state $i$ to state $j$.

For the first problem, there are two states: a person will be either in city A (state 1) or in city B (state 2). If you're in state 1, the probability of switching to state 2 is $.04$, and the probability of staying to state $1$ ("moving from state 1 to state 1") is $0.96$. So, $P_{11} = 0.96$ and $P_{12} = 0.04$.

Similarly, $P_{21} = 0.08$ and $P_{22} = 0.92$. All together, we have $$ P = \pmatrix{0.96 & 0.04 \\0.08 & 0.92} $$ For the second problem, we have three states: state 1 is "in city A". State 2 is "in city B". State 3 is "away from the coast for good". See if you can figure out what the transition matrix should look like. Hint: $P_{33} = 1$.

Answer 2

You're pretty much right for (a). For (b), consider having a three-state system that includes "Lives in A", "Lives in B", "Lives in neither". Then there's a transition from "Lives in A" to "Lives in neither" of 0.02, and one from "Lives in B" to "Lives in neither" of 0.04, while the transitions for "Lives in neither" to the three states is 0, 0, 1 (since it's implied that no-one new ever moves back to the coast).

Then, once you've got that transition matrix, just remove the "Lives in neither" parts, because the loss of people from the coast is actually already handled by the fact that the transitions between A and B don't sum to 1 any more.