4. A population starts off with just one female. Each female lives for a fixed time I and then dies. The number of female off springs of a single female is given by the following probability distribution x 0 1 pr(x) 0.2 0.4 2 3 0.3 0.1 i) Find the probability distribution of the second generation. ii) Determine the probability of extinction by second generation. iii) Determine the probability of extinction at second generation. iv) Determine the probability of extinction by third generation. v) Different females are independent. Find the probability of ultimate extinction. 5. A population starts off with just one female. Each female lives for a fixed time T and then dies. The number of female off springs of a single female is given by the following probability distribution X 0 1 3 i) ii) iv) v) pr(x) 0.4 0.2 2 0.3 0.1 Find the probability distribution of the second generation. Determine the probability of extinction by second generation. Determine the probability of extinction at second generation. Determine the probability of extinction by third generation. Different females are independent. Find the probability of ultimate extinction. 6. Ghosts reproduce according to a branching process with family size distribution Y Binomial (2, 0.7). i) ii) BE iii) Working directly from the probability function, show that the PGF of Y is G(s) (0.7s+0.3) Find the probability that a family of ghosts will eventually go extinct. Suppose there are 8 ghosts alive at generation 12. What is the probability of eventual extinction? 7. A population reproduce according to a branching process with family size distribution Y ~ Binomial (3, 0.5). Calculate the a) distribution of the second distribution b) probability of extinction by third generation c) probability of extinction at third generation d) probability of ultimate extinction. e) mean of ninth generation