5.2 Simple Death Process

Consider a population of individuals which is subject to random death. Assume probability of any individual dying in time \(\Delta t\) is \(\mu \Delta t\).
Assume individuals dies independently and there are \(a\) individuals initially. Let \(X(t)\) denote the size of the population at time \(t\). Then the chance of one death in interval \(\Delta t\) is \(\mu \, X(t)\, \Delta t\).

Let \(P_n(t) = P\left (X(t) = n\right )\) \[P_n(t + \Delta t) = P_n(t)\, \left (1 - n\, \mu \, \Delta t\right ) + P_{n + 1}(t)(n+1) \mu \Delta t\, , \hspace {0.3cm} n\leq a\]

\[\frac {P_n(t + \Delta t) - P_n(t)}{\Delta t} = -n\, \mu \, P_n(t) + (n + 1)\, \mu \, P_{n + 1}(t).\] As \(\, \Delta t \longrightarrow 0\), \begin {equation} P'_n(t) = -n \, \mu \, P_n(t) + (n + 1)\, \mu \, P_{n + 1}(t)\, ,\hspace {0.3cm} 0\leq n\leq a. \end {equation}

Let \(M(t,\theta )\) be the mgf of \(X(t)\). \[M(t,\theta ) = \sum ^a_{n = 0} e^{n\theta }\, P_n(t)\]

\[\frac {\partial M}{\partial \theta } = \sum ^a_{n = 0} n\, e^{n\theta }\, P_n(t)\] \[\frac {\partial M}{\partial t} = \sum ^a_{n = 0} e^{n\theta }\, P'_n(t).\]

\[X(0) = a\, \implies \, P_a(0) = 1\, \implies \, M(0,\theta ) = e^{a\theta }.\] Multiply equation (5.1) by \(e^{n\theta }\) and sum for \(n\) from 0 to \(a\) \[\sum ^a_{n = 0} e^{n\theta }\, P'_n(t) = -\mu \, \sum ^a_{n = 0}n\, e^{n\theta }\, P_n(t) + \mu \, \sum ^a_{n = 0} (n + 1)\, e^{n\theta }\, P_{n + 1}(t).\]

\[\frac {\partial M}{\partial t} = -\mu \, \frac {\partial M}{\partial \theta } + \mu \, e^{-\theta }\, \frac {\partial M}{\partial \theta }.\] \[\frac {\partial M}{\partial t} \, + \, \mu \left (1 - e^{-\theta }\right )\, \frac {\partial M}{\partial \theta } = 0.\]

\[\frac {dt}{1}\, =\, \frac {d\theta }{\mu \left (1 - e^{-\theta }\right )}\, = \, \frac {dM}{0}\] \(dM = 0\, \implies \, M = \)constant.

\[\mu \, dt = \frac {e^{\theta }}{e^{\theta } - 1}\, d\theta \] \[A + \mu t = \ln \left (e^{\theta } - 1\right )\] \[e^{\theta } - 1 = e^{A + \mu t}\] \[e^A = e^{-\mu t}\, \left (e^{\theta } - 1\right ).\] Hence a general solution of the PDF is \[M(t,\theta ) = \Psi \left (e^{-\mu t}\, \left (e^{\theta } - 1\right )\right ).\] Since \(M(0,\theta ) = e^{a\theta }\), we have \[\Psi \left (e^{\theta } - 1\right ) = e^{a\theta }.\]

Let \(e^{\theta } - 1 = u\) \[e^{\theta } = u + 1\] \[e^{a\theta } = \left (u + 1\right )^a\]

\[\Psi (u) = \left (u + 1\right )^a\] Hence \[M(t,\theta ) = \left [1 + e^{-\mu t}\, \left (e^{\theta } - 1\right )\right ]^a.\]

\[P(t,\theta ) = \left (1 + e^{-\mu t}\, \left (\theta - 1\right )\right )^a.\]

\(Y:\) a binomial random variable \((n,p)\)
\(P(t):\) probability generating function of \(Y\)

\begin {align*} P(t) & = \sum ^n_{r = 0} t^r\, Pr(t) = \sum ^n_{r = 0} t^r\, C_r^n\, p^r\, q^{n - r}\\ & =\sum ^n_{r = 0} C^n_r\, (tp)^r\, q^{n - r}\\ & = (tp + q)^n\\ & = (1 - P + tp)^n\\ & = (1 + p(t - 1))^n. \end {align*}

\(X(t)\) has Binomial distribution with parameters \(a\) and \(e^{-\mu t}\).
Hence \[E(X(t)) = a\, e^{-\mu t}\] As \(\, t \longrightarrow \infty \), \(E(X(t)) \longrightarrow 0\). i.e eventually the population will die out.

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