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limitation distribution in markov chain

Let Zn denotes the number of a particular type of virus in generation n. At generation 0, there was just one virus, or in other words Z0 = 1. The i-th virus in generation n has Xi,n many offsprings, where n = 0,1,2,..., and i = 1,2,...,Zn. Let Xi,n’s are i.i.d Poisson(λ) random variable for all n = 0,1,2,..., and i = 1,2,...,Zn.

(a) (3 pts) Compute E(Zn).

(b) (2 pts) Let λ = 0.5. Compute limn→∞ E(Zn).

(c) (3 pts) Show that Zn is a Markov Chain. What is the state space of this MC?

(d) (2 pts) What are the absorbing states in this MC? (2 pts)

You may use the following hint: For n ≥ 0, Zn+1 can be written in terms of Zn, and Xi,n’s for i = 1,2,...,Zn.

Kĩ năng: Phân tích thống kê, Toán học

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