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## Homework Statement

Consider a discrete time birth and death process in which the maximal population

size is N = 6. Birth rates and death rates are directly proportional to the current

size Xt of the population at time t (t = 0; 1; 2; : : :). If the maximal population size

is reached, no more births can take place. That is bi = bi (i = 0; 1; : : : ; 5); b6 = 0

di = di (i = 0; 1; : : : ; 6) At each time step, either one birth or one death or neither (but never both) will take place.

Suppose the population starts with size 3 (i.e., X0 = 3). Let b = d = 0.02.

Write down the transition probability matrix of the Markov chain.

## Homework Equations

All probabilities in the matrix must be 0<=x<=1 and must sum up to 1 across the rows.

Possible states of the birth and death process are the possible sizes of the population (current state = current population size).

Also, the growth rate of the population through births is proportional to the population size with proportionality constant λ as long as the population remains below N (here N=6). The death rate is also proportional to the size of the population with proportionality constant μ as long as the population size remains strictly positive. That is bi = λi and di=μi for λ,μ >0 and 0<I<N

## The Attempt at a Solution

I'm having a problem setting this up. It seems to me that if the population starts with 3, then the first 3 rows should all be 0 since if the population starts at 3, then the first 3 states shouldn't really matter initally. Here is what I have come up with.

0 1 2 3 4 5 6

0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

2 0 0 0 0 0 0 0

3 0 0 .06 .88 .06 0 0

4 0 0 0 .08 .84 .08 0

5 0 0 0 0 .10 .80 .10

6 0 0 0 0 0 .12 .88

For states 3-6, I have multiplied the current state * the birth or death rate to get the probability of going to the next or previous state.

Please help.

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