Hello, I'm learning stochastic processes currently, and I'm very stuck on a certain problem of my homework. I've tried a few things, but I don't know where to start. Any help is greatly appreciated. Thank you in advance for your help!

1. In a family, denote Xn as the number of individuals in the nth generation. Each individual independently reproduoe into a random number of offspring who live in the next generation. This random number is I} with probability p2, 1 with probability 2p[1 p}, and 2 with probability [1 pf. The parameter p is within the range of {I}, 1,32). a.) Let p = 1K3. 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 lllI individuals that shares the same reproducing rule? (4 points} b.) Let p = If 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 T be exact number of generation when the family extincts. Given that the rst generation has 1 individual, compute Pf'r = 2] and P(T = 3). {4 points) d.) Let p = 2X3. 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]