Problem 1. Consider a randomly growing population in which there are Zn individuals in thepopulation at generation n for n = 0, 1, 2, . . .. Each individual in generation n gives birth to a randomnumber of offspring: say the ith individual in the nth generation gives birth to X(n)i individuals ingeneration n + 1, where {X(n)i : i 1, n 0} are independent and identically distributed randomvariables, each with range {0, 1, 2, . . .}. Then we can define Zn recursively:Zn+1 =(X(n)1 + · · · + X(n)k if Zn = k for k 10 if Zn = 0,and we suppose the population starts with Z0 = 1, a single individual at generation 0. Suppose themean number of offspring per individual in the population is E(X(n)i ) = .a) Compute then numbers E[Zn+1|Zn = k] for k 0.b) Express E[Zn+1|Zn] as an expression of the R.V. Zn.c) Compute E[Zn].d) Suppose that 0 < < 1. Use Markovs inequality to show that limn P (Zn = 0) = 0. Whatdoes this mean for the population?