Discussion
Consider a branching process \(\{X_{n}, n \ge 0\}\). Let \(X_{0} = 1\) and assume that \(E\left(\sum_{\theta}^{\infty} X_{n}\right) \lt \infty\). Then, the extinction probability :
Answer: D
The extinction probability equals 1.
The sum is printed from \(\theta\), meaning from generation 0. With mean offspring \(m\), \(E(X_{n}) = m^{n}\) when \(X_{0} = 1\), so
\[ E\left(\sum_{n} X_{n}\right) = \sum_{n} m^{n} \]
which is finite only if \(m \lt 1\). A subcritical branching process dies out with probability 1. (Equivalently, a finite expected total progeny forces the total progeny to be finite almost surely, so the population cannot survive forever.)
An extinction probability in (0, 1) arises only for \(m \gt 1\), when the expected total is infinite; 0 is impossible once there is any chance of having no offspring, and the extinction probability always exists as a limit of \(P(X_{n} = 0)\).
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