Consider a branching process $(Xn)_{n≥0}$ but with the offspring distribution given by $Z ∼ NegBinomial(M, p)$ where $P(Z = n)$ $ ={n+M-1 \choose n}$$(1-p)^M$$(p^n)$
Find the range of values for $p$ such that the population becomes extinct with probability $1$.
I am unsure of how to proceed from here. Any help appreciated
The extinction probability is $1$ exactly if the expected number of offspring of an individual is $\le1$ (unless every individual always has exactly $1$ offspring).
The expected number of offspring in this case is $\frac{pM}{1-p}$. From $\frac{pM}{1-p}\le1$ we obtain $p\le\frac1{M+1}$.