A migration model. Consider the following simple model of an animal population in an open habitat. Each animal living there leaves the habitat, independently of all others, with probability 1  p, and it stays with probability p. At the same time, a Poisson number (with parameter a > 0) of animals immigrates from the outside world.

(a) Describe the number Xn of animals living in the habitat by a Markov chain and find the transition matrix ….

(b) Calculate the distribution of Xn when the initial distribution is P, the Poisson distribution with parameter > 0.

(c) Determine a reversible distribution ˛.