A model for reproduction works as follows. A population starts with a single individual. At the end of one generation, this This type of model is relevant in many and varied scenarios. The population might consist of animals, or diseased cells in a tumour, or a computer virus that aims to spread itself. It could even describe a rumour being passed around.

individual spawns some children, and then it immediately dies.

The number of children it produces is a random variable, X.

The model is known as a "branching process", because we often picture the population like a family tree, branching at every generation.

Each of the children live for one generation, then they each have their own children according to the same distribution X, and die themselves. The process goes on and on for ever, or until there are no individuals left. If there are no children produced at the end of a generation, the population is said to have become extinct.

Suppose that the number of children produced by each individual is X ∼ Geometric(0.5).

a. Under this model, what is the probability that any given individual has no children? What is the probability it has four or more children?

b. Sketch the probability function of the number of children produced by any individual.

c. What is the probability that the population becomes extinct at the end of the first generation?

d. Remarkably, for this model we can calculate the distribution of the number of individuals alive in any generation.
Let Zn be the number alive at the end of generation n, for n = 1, 2, . . .. The probability function of Zn is Remember all individuals die as soon as they have children, so the number alive at the end of any generation is the same as the number just born.
P(Zn = 0) = n n + 1 ;
P(Zn = z) = 1 n(n + 1)

n n + 1  z for z = 1, 2, 3, . . .
Use this formula to calculate the probability that the population is extinct at the end of generation 1. Check that you get the same answer as you did in part (c).
Theory tells us that this population is guaranteed to go extinct eventually, but it can take a long time. After 10 generations you should find there is a high probability of extinction, but there is also a reasonable probability of having a few individuals alive and kicking.

e. Find the probability function and CDF of the number born at the end of generation 10, tabulating each function as far as z = 2. Round all probabilities to 5 decimal places.

f. Find the probability that, at the end of generation 10:
i. the population is extinct.
ii. there are at least three individuals alive.