Problem 5: lconsider a simple GaltonWatson branching process for a population of {normal} stem cells with the following parameters: I The population starts from one cell : Ni} = 1. a At each generation: each stem cell can either die with probability % or divide into two stem cells with probability %. Simulate this process from generation t = Ill to generation t = l. o Compare the analytical expression of the expected value of the size of population at each generation t with the value estimated with simulations. o Do the same thing for the probability of extinction of the population by con sidetingt=1lilast=oo Using the recursive equation between the MGF F [t + 1,2} of NIH + 1} and the MGF F{t,z} of NH}: I Derive a method to compute P[N[t]l :- U}. o Compare the computed probability with the values estimated from simulations for t = 15