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 mn be the largest possible number of bacteria at time n. Find mn and compute the probability that there are exactly mn 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?