Consider the following variation of the branching process. At
each time n, each individual produces offspring independently using
offspring distribution {pk}, and then the individual dies with
probability q (0,1). Hence, each individual reproduces j times
where j is the lifetime of the individual. For which values of q,
{pk} do we have eventual extinction with a probability of one?
(Hint: the answer is quite clean, at the end it should relate a
common statistic of {pk} to q. Be clear and show your
work.)