

\subsection{\bf Basic Birth Process Formalism---II}

\begin{slide}
\begin{itemize}
\item Then, the probability of $k$ events at time $t+dt$ is
\[
P_k(t+dt) = P_k(t)[1-\lambda dt] + P_{k-1}(t) \lambda dt + o(dt)
\] 

\item or
\[
\frac{dP_k}{dt} (t) = -\lambda P_k(t) + \lambda P_{k-1} (t)
\hspace{1em} :k\ne 0
\]

\item For $k=0$, the above is modified as $P_{0-1}$ term doesn't exist, 
i.e.,
\[
\frac{dP_0(t)}{dt} = -\lambda P_0(t) \hspace{1em} : k=0
\]

\item Solving the last gives $P_0(t) = e^{-\lambda t}$ and then 
induction shows
\[
P_k(t) = e^{-\lambda t} \frac{(\lambda t)^k}{k!}
\]
and we have achieved Poisson again.  

\item It is now obvious why any counting experiments for random events
should have a Poisson Distribution.
\end{itemize}
\end{slide}

