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What is the expected long term behaviour of a population that dies out with probability $0.2$, stays the same with probability $0.5$ and doubles with probability $0.3$?

Of course, $$ E[X_{n+1}] = 0.5X_n + 0.3\cdot2X_n = 1.1X_n $$ and thus the expectated population size goes to infinity for $n\to\infty$.

But the hint given for the exercise was that this is is classic Markov chain problem that can be solved easily with a transition probability matrix. I don't really see how that could make the solution easier or did I get the question wrong?

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    $\begingroup$ Are you sure that it's the whole population that either all dies out, doubles or stays the same, or each individual in the population that either dies out, doubles or stays the same (independently of each other)? Your analysis as it is stands anyway because of linearity of expectation, and is probably the easiest approach. $\endgroup$
    – jlammy
    Commented Jan 23, 2021 at 16:32

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Assume the population size at time $0$ is $P_0$.
After 1 unit of time, the size will be either $0,P_0$ or $2P_0$.
After $N$ units of time, the population will be one of $0,P_0,2 P_0,...,2^N P_0$.
Take $N$ very large but fixed and assume those are the possible states and find the probability of being in each state at each time using the transition matrix.
It is possible to directly calculate the probabilities, but it is a good exercise to try to do it with the transition matrix and that is what you are expected to do from the hint.
The probability of never reaching $0$ is $0.8^N$, hence the probability of reaching $0$ is $1-0.8^N$.
The probability of having size $2^k P_0$ is ${{N}\choose{k}}0.5^{N-k} 0.3^k$.
The expected value can be calculated directly from there and it is equal to $1.1^N P_0$.
You can check for the case $N=1$ that the expected value is $0.5 P_0+0.3 (2) P_0=1.1 P_0$ and use induction for the general case.

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The transition matrix A, for the states: stable, double, die out

| 0.5 | 0.3 | 0.2|

| 0.5 | 0.3 | 0.2|

| 0 | 0 | 1 |

A population that died out can't increase. The long term behavior is given by the stationary distribution pi.

pi A = pi

The result is that pi = [0 0 1]. Thus the long term behavior is that the population goes extinct.

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