5.2 The Simple Death Process
Each individual independently dies in \((t, t + h]\) with probability \(\mu h + o(h)\), and there are no births. With \(X(0) = a\) and \(n\) alive, a death occurs with probability \(n\mu h + o(h)\), so \begin {equation} \label {eq:deathde} P_n'(t) = -n\mu P_n(t) + (n + 1)\mu P_{n+1}(t),\qquad 0\leq n\leq a . \end {equation}
Theorem 5.2.1. For the simple death process with \(X(0) = a\), \(X(t)\) is binomial with parameters \(a\) and \(e^{-\mu t}\): \[P(X(t) = n) = \binom an e^{-n\mu t}\left (1 - e^{-\mu t}\right )^{a - n},\qquad n = 0, 1, \dots , a .\]
Proof. As before, the m.g.f. satisfies \(\frac {\partial M}{\partial t} + \mu (1 - e^{-\theta })\frac {\partial M}{\partial \theta } = 0\) with \(M(\theta , 0) = e^{a\theta }\). Along the characteristics, \(e^{-\mu t}(e^\theta - 1)\) is constant, so \(M = \Psi \left (e^{-\mu t}(e^\theta - 1)\right )\) with \(\Psi (u) = (1 + u)^a\), and \[M(\theta , t) = \left [1 + e^{-\mu t}(e^\theta - 1)\right ]^a = \left [(1 - e^{-\mu t}) + e^{-\mu t}e^\theta \right ]^a,\] the m.g.f. of a binomial distribution with \(a\) trials and success probability \(e^{-\mu t}\). □
The result can be seen directly: each of the \(a\) individuals has an exponential lifetime with rate \(\mu \), so is still alive at time \(t\) with probability \(e^{-\mu t}\), independently of the others. The expected size \(ae^{-\mu t}\) tends to zero, and the population is sure to die out.
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