(Answered)-Suppose that we have a branching process where the offspring - (2025 Updated Original AI-Free Solution

Question

1. Suppose that we have a branching process where the offspring distribution is different in the even and odd generations. Let X,Y be discrete random variables with non-negative integer values. Assume that we start the process with a single individual and we grow a family tree with the following rules: ? In the first step the individual has kids according to the distribution of X.
   ? After step n the individuals in the last generation all have kids independently according to the distribution of X if n is even and according to the distribution of Y if n is odd. PLEASE Express the probability generating function of the size of the nth generation using the probability generating functions of X and Y .