Hello,

I have been reading a few things about Markov chains and I want to clarify a few things using a made-up problem as an example. Any help would be appreciated greatly.

A group of scientists create an artificial lake to study the lifespan of a certain type of fish. The scientists populate a transition matrix describing the movements of the fish between one length division to another length division at the end of each month.

Lets call the states:

A: 10-20cm

B: 21-30cm

C: 31-40cm

D: 41-50cm

and include a final state E: denoting the fish has been removed from the system because the scientists got hungry and went fishing, or the fish died of natural causes. (Also, due to some chemical in the water, the fish cannot breed).

Let $\displaystyle P = \begin{pmatrix}0.3 & 0.3 & 0.2 & 0.1 & 0.1 \\ 0 & 0.4 & 0.2 & 0.2 & 0.2 \\ 0 & 0 & 0.4 & 0.3 & 0.3 \\ 0 & 0 & 0 & 0.4 & 0.6 \end{pmatrix}$ where the states read A, B, C, D, E across the top.

Okay, now here are my questions:

(1) Would these probabilities represent that a randomly selected fish in state A in the current month, will move to state D in the next month with probability 0.1?

(2) Lets say the current population on the 30th of September in each state is as follows:

A: 500 fish

B: 400 fish

C: 300 fish

D: 100 fish

E: 0 fish

Can I use this transition matrix to predict/calculate (and if so how):

(i) the probability that there will be 500 fish in state B at the end of October?

(ii) The probability that by the end of October there will be 500 fish in state B and 400 fish in state C?

(iii) The number of months it would take for all the fish population to die (state E)?

(iv) The number of months it would take to have 200 fish in state D and 380 fish in state C?

and finally

(v) what would the population look like in 3 months(in terms of the overall fish distribution across the states)?

I am really not sure if the last two are possible to predict from the matrix.

If anyone could provide me with any insight, you would make me very happy.

Thanks