I'm doing some work on branching processes and would like to know where the process becomes extinct. If $X$ is the number of offspring of an individual, then the process goes extinct when $\mathbb{E}[X] \leq 1$.
I know that the probability that each cell in the first generation splits into $k$ individuals is $qp^k$, where $k \geq 0, p + q = 1$.
For which values of the parameters $p,q$ does the process become extinct?
The answer says that the process becomes extinct if the mean number of offsprings is $\leq 1$. It then shows that $q \displaystyle\sum^\infty_{k=0}kp^k = \frac{p}{q}$.
Why do we start with $q \displaystyle\sum^\infty_{k=0}kp^k$? What does this have to do with the mean or the expected value?
The number of offspring of a cell is $X$.
The probability that $X=k$ is given by: $~\mathsf P(X=k) = qp^k$ for $k\in\{0, 1, 2,\ldots \}$ and $q=(1-p)$.
The expected value of $X$ is given by : $~\mathsf E(X) = \sum_{k=0}^\infty k~\mathsf P(X=k)$
Thence, using Geometric Series closed form (when $\lvert p\rvert<1)$: $$\begin{align}\mathsf E(X) ~=~& \sum_{k=0}^\infty k~qp^k \\ ~=~ & (1-p)\sum_{k=0}^\infty kp^k \\ ~=~ & \dfrac{(1-p)p}{(1-p)^2} \\ ~=~ & \dfrac{p}{q} \end{align}$$
That is all.