At time 0 . a blood culture starts with one red cell. At the end of one minute, the red cell dies and is replaced by the following combinations with probabilities as indicated: 7: 2 red ces % 1 red, 1 White i 2 White cells. 12 Each red cell lives for one minute and then gives birth to offspring in the same way as the parent cell. Each white cell lives for one minute and dies without reproducing The individual cells act independently of each other. Let X" denote the number of red cells attime n + 1/2. 1. Explain why {X,n > 0} is a branching process with Xo = 1 and the offspring distribution: fo = 1/12, f1 = 2/3, f2 = 1/4. 2. Calculate the probability of the event that there are no red cells at time 2.5. 3. Find the expected number of red cells n minutes after the culture began and the probability that the culture eventually dies out. 4. Suppose that, at time 0, a blood culture produces red cells according to a Poisson distribution with mean A = 2 rather than just one red cell (other facts related to this question do not change). Find the expected number of red cells n minutes after the culture began and the probability that the culture eventually dies out