

\begin{slide}
\subsection{\bf General Birth and Death Process}

\begin{itemize}
\item Consider (Feller, p. 404) a system with a denumerable number of
states $E_n$ whose probability of occupation varies with $t$.

\item For Poisson $E_n$ is ``we have observed $n$ events.'' 

\begin{description}
\item{$\bullet$} Note only one $E_n$ occupied at a time 
\item{$\bullet$} and $\exists$ chance $\lambda dt$ of a transition $E_n
\rightarrow E_{n+1}$.
\item{$\bullet$} This is a pure birth process.
\end{description}

\item We can generalize $\lambda$ to $\lambda_n(t)$:  a function of $n$
and $t$.

\item Also, we can have death processes which also allow relapses $E_n$
to $E_{n-1}$ with chance $\mu_n(t) dt$.  So:

\begin{tabular}{cclcr}
$\bullet$ & change & $E_n \rightarrow E_{n+1}$ & chance & $\lambda_n dt
+ o(dt)$ \\
$\bullet$ & & $E_n \rightarrow E_{n-1}$ & chance & $\mu_n dt + o(dt)$ \\
$\bullet$ & Any other & change & chance & $o(dt)$
\end{tabular}

\item We can, at once, derive a similar differential equation for $P_n
(t)$:  the probability of system being in state $E_n$.
\begin{eqnarray*}
\frac{dP_n (t)}{dt} = & - & \left(\lambda_n + \mu_n \right) P_n (t) \\
& + & \lambda_{n-1} P_{n-1} (t) + \mu_{n+1} P_{n+1} (t)
\end{eqnarray*}

\item with $\mu_0 = 0$ and $\lambda_{-1} = 0$ so that above is still
correct with $n=0$.
\end{itemize}
\end{slide}

