1. In a family, denote X\" as the number of individuals in the nth generation. Each individual independently reproduce into a random number of oifspring who live in the next generation. This random number is 0 with probability 132, 1 with probability 2p(1 p), and 2 with probability (1 p)2. The parameter p is within the range of (0, 1 / 2). a.) Let p = 1/3. Suppose that the family starts from 2 individuals, how much larger is the extinction probability compared to the extinction probability for a family starting frOm 10 individuals that shares the same reproducing rule? (4 points) b.) Let p = 1/3. If the number of individuals in the rst generation follows a Poisson distribution with mean parameter 2, what is the extinction probability for this family? IS this probability larger than the extinction probability for a family that starts with exactly 2 individuals? (4 points) c.) Consider a general value of p. Let 7' be exact number of generation when the family extincts. Given that the rst generation has 1 individual, compute P('T = 2) and P(7' = 3). (4 points) (:1) Let p = 2/3. The family starts with one individual. What is the expected number of individuals that ever exist in this pOpulation? What if the family starts with two individuals? (4 points)