(solution) Provide all details of this last proof. Hence, the expected number of members in a branching…

(solution) Provide all details of this last proof. Hence, the expected number of members in a branching…

Provide all details of this last proof. Hence, the expected number of members in a branching process grows or decays exponentially fast. If the expected numebr of children is larger than 1, the expection grows to infinity, if it is smaller, it decays to zero. This fact should make us curious as to whether it is possible that the branching process survives forever.To this end we first prove: