Consider the following model for growth of a population. Each individual in the population lives for exactly one 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' offspring with probability 'a_i',i=0,1,2, independent of the other individuals in teh population. Let the size of the population at the beginning of the nth time period (that is, the size of the nth generation) be denoted by Xn. Suppose the population begins at time 0 with a single individual.
a. 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 to the next generation).
b. Classify the states of the Markov chain
c. Without calculating, argue what the long run population size will be.