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{\bf Question}

Describe the characteristic properties of the Poisson process. Let
$X(t)$ denote the total number of events occurring in a time
interval of length $t$ in a compound Poisson process. Then $X(t)$
has probability generating function

$$G_t9z)=\exp(\lambda t A(z)-\lambda t).$$

Explain what is meant by a compound Poisson process, and the
meaning of $\lambda$ and $A(z)$ in the formula for $G_t(z)$.

${}$

Coachloads of student visit the Hirst memorial museum every day.
It is open from 8 a.m. to 6 p.m. Coaches arrive according to a
Poisson process of rate $\lambda$ an hour, and each coach carries
a full load of $n$ passengers. Each visitor decides independently
with probability $p$ to contribute $£1$ towards upkeep of the
museum. Find the mean and variance of the total amount of money
contributed each day.


\vspace{.25in}

{\bf Answer}

Let $N(a,\ b)$ denote the number of events occurring in the time
interval $a < t \leq b$. This is an integer-valued random
variable. These random variables form a Poisson process if the
following properties hold.

\begin{description}
\item[(i)]
Numbers of events in non-overlapping intervals are independent.

\item[(ii)]
The probabilities of an event occurring in a small time interval
is roughly proportional to its length.

i.e.

$P(N(t,\ t+\delta t)=1)=\lambda \delta t+o(\delta t)$

$P(N(t,\ t+\delta t)=0)=1-\lambda \delta t+o(\delta t)$

as $\delta t \to 0$.

\item[(iii)]
$\lambda$ is constant i.e. we have time homogeneity.
\end{description}

$(i)$ and $(ii)$ is equivalent to saying that the number of events
occurring in a time interval of length $t$ has a Poisson
distribution with parameter $\lambda t$.

${}$

Now suppose that

\begin{description}
\item[(a)]
points occur in a Poisson process with rate $\lambda$. Let $N(t)$
be the number of events in time $t$.

\item[(b)]
$Y_i$ events occur at each point, where the $Y_i$ are i.i.d random
variables

\item[(c)]
$Y_1,\ Y_2, \cdots$ and $N(t)$ are independent.

\end{description}

The total number of events occurring in time $t$ is
$$X(t)=\ds\sum_{i=1}^{N(t)}Y_i$$

and is said to have a compound Poisson distribution. If each $Y_i$
has probability generating function $A(z)$ then $X(t)$ has
probability generating function

$$G_t(z)=\exp(\lambda t A(z)-\lambda t)$$

${}$

In this example the number of people in a coachload who decide to
contribute will be a $B(n\, p)$ random variable, with probability
generating function $(q+pz)^n$, where $q=1-p$.

${}$

The museum is open for 10 hours, and so the probability generating
function for the total contribution is

$$G(z)=\exp(10\lambda(q+pz)^n-10\lambda)$$

The mean is given by $G'(1)$ and the variance by
$G''(1)+G'(1)-G'(1)^2$.

\begin{eqnarray*} G(z) & = & \exp(-10\lambda)\cdot
\exp(10\lambda(q+pz)^n)\\ G'(z) & = & \exp(-10\lambda)\cdot
\exp(10\lambda(q+pz)^n)\cdot 10 \lambda n p (q+pz)^{n-1}\\
\rm{so}\ G'(1) & = & 10\lambda n p\ \rm{since}\ p+q=1\\ G''(z) & =
& \exp(-10\lambda)\cdot\exp(10\lambda(q+pz)^n)\cdot
100\lambda^2n^2p^2(q+pz)^{2(n-1)}\\ & & +
\exp(-10\lambda)\exp(10\lambda(q+pz)^n)\cdot\\ & & \ \ 10\lambda
n(n-1)p^2(q+pz)^{n-2}\\ G''(1) & = & 100\lambda^2n^2p^2+10\lambda
n(n-1)p^2\\ \rm{so}\ & & G''(1)+G'(1)-G'(1)^2\\ & = &
100\lambda^2n^2p^2+10\lambda n(n-1)p^2+10\lambda n p -(10\lambda
np)^2\\ & = & 10 \lambda n p ((n-1)p+1)\\ & = & 10\lambda
np(np+q)\end{eqnarray*}




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