Consider the following model for growth of a population. Each individual in the population lives exactly on time unit (so think of time units as generations) and has the ability to produce offspring of the same kind at the end of its lifetime. Each individual produces i offsprings with probability \alpha(sub i), i = 0,1,2, independent of the other individuals in the population. Let the sie of the population at the beginning of the n^th timeperiod (that is, the size of the n^th generation) be denoted by Xn. Suppose the population begins at time 0 with a single individual.
a) Justify that {Xn, n = 0,1,2...} is a MArkiv chain.
b) Determine the one-step transition probabilities for this Markov chain (Hint: since each individual produces offspring independently of the others, think about adding up the offspring of each individual to get the next generation).
c) Classify the states of the Markov chain.
d) Without calculating, argue what the long run probability size will be.