Question:
A pure death process $\{X(t); t \ge 0\}$, where $X(t)$ denotes the number of individuals alive at time $t$, starts with $X(0) = 8$. The lifetime of each of these individuals is exponential with mean $\frac 1 \upsilon$.
Solve for the median time until the population dies out.
Attempt:
Let $T_n$ be the time between the $(n-1)$th death and the $n$th death where $n = 1,2,\dots,8$. Let $W_n = T_1 + T_2 + ... + T_n$. Then $P(W_n \le t) = P(X(t) \ge n)$, so $P(W_8 \le t) = P(X(t) \le 8-n) = P(X(t) = 0)$.
Let $t_m$ be the median time till population dies out. Then $P(W_8 \le t_m) = 0.5$ so $P(X(t_m) = 0) = 0.5$.
Since $X(t)$ is distributed as $Bin(8, \Bbb e^{-\upsilon t})$, then $P(X(t_m) = 0) = (1-exp(- \upsilon t_m))^8 = 0.5$, so $t_m \approx \frac {\ 2.489} {\upsilon}$.
