. Bacteria reproduce by cell division. In a unit of time, a bacterium will either die (with probability ), stay the same (with probability ), or split into 2 parts (with probability ). The population starts with 100 bacteria at time n = 0.

1.Write down the expression for the generating function of the distribution of the size of thepopulation at time n N0. (Note: you can use n-fold composition of functions.)

2.Compute the extinction probability for the population.

3.Let m_n be the largest possible number of bacteria at time n. Find m_n and compute the probability that there are exactly m_n bacteria in the population at time n, n N0.

4.Given that there are 1000 bacteria in the population at time 50, what is the expected number of bacteria at time 51?