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\begin{center}
MA181 INTRODUCTION TO STATISTICAL MODELLING
THE PROBABILITY GENERATING FUNCTION
\end{center}
\begin{description}
\item[Definition]
Suppose $X$ is a discrete random variable that can take only
non-negative integer values, i.e.$X$ has range $R$ that is a
subset of $\{0,1,2,\ldots\}$; $X$ might be the number of piglets
in a litter or the number of letters you receive in the post
today. In fact, the large majority of discrete random variables
met in practice have a range of this type. If $p(x)$ is the
probability function of $X$, consider the function $G(s)$ defined
by
\begin{equation}
G(s)=\sum_{x\in R}s^xp(x)=E(s^X),
\end{equation}
where the real variable $s$ is a mathematical variable, as opposed
to a random one, and, in a context such as this, is described as a
\textit{dummy variable} since it has no interpretation in terms of
the problem being discussed. In order to see the use to which
$G(s)$ might be put, consider a single example.
\item[Example 1]
A builder has completed a small development of three houses, and,
from all the available evidence, the number $X$ of then that he
will sell within six months has the distribution given by
\begin{center}
\begin{tabular}{ccccc}
\hline $x$&0&1&2&3\\
$p(x)$&$\frac{1}{16}$&$\frac{5}{16}$&$\frac{7}{16}$&$\frac{3}{16}$\\
\hline
\end{tabular}
\end{center}
Since $X$ is a discrete random variable with an appropriate range,
the function $G(s)$, defined at (1), is given by
\begin{eqnarray*}
G(s)&=&\sum_{x=0}^3s^xp(x)\\ &=&(s^0+5s^1+7s^2+3s^3)/16\\
&=&(1+s)^2(1+3s)/16.
\end{eqnarray*}
Thus $G(s)$ can be expressed as a factorised polynomial in $S$.
Since a basic grounding in mathematics equips us to handle such
expressions with relative ease, we should not be surprised to
learn that, within this function, we have a convenient method for
manipulating the distribution of $X$ and deriving its properties.
Note that, for any distribution, $G(1)=1$.
\item[Inversion of $G(s)$]
So much for the definition of $G(s)$; suppose now we are given a
$G(s)$ like, for instance, the one derived in Example 1. By
expanding this as a polynomial or power series in $s$, and picking
out the coefficients, we can reconstruct the probability function
of $X$; the value of $p(x)$ is found as the coefficient of $s^x$.
For this reason, the function $G(s)$, defined at (1), is called
the \textit{probability generating function} (pgf) of $X$.
\item[Example 2]
If $G(s)=(4-3s)^{-1}$ is the pgf of a random variable, what is its
distribution? Note that $G(1)=1$. Expanding $G(s)$ as a power
series in $s$, we obtain
$$G(s)=\frac{1}{4}\left[1-\left(\frac{3s}{4}\right)\right]^{-1}=
\frac{1}{4}\left[1+\left(\frac{3s}{4}\right)+\left(\frac{3s}{4}\right)^2+\ldots\right],$$
all the coefficients of which lie in $[0,1]$. Calling the random
variable $X$, we can find its probability function by extracting
these coefficients. The result can be summarised by the formula
$$p(x)=\frac{1}{4}\left(\frac{3}{4}\right)^x,\ x=0,1,2,\ldots$$
\item[Moments]
Consider next the derivatives of $G(s)$ with respect to $s$. The
first derivative is, from (1),
$$G'(s)=\frac{\partial G(s)}{\partial s}=\sum_xs^{x-1}xp(x).$$
Setting $s=1$ in this expression leads to
\begin{equation}
G'(1)=\sum_xxp(x)=E(X).
\end{equation}
Hence, if we know the pgf of a distribution, we can derive its
mean by the use of this result. Note carefully the meaning of
$G'(1)$: the pgf is first differentiated with respect to $s$, and
only after that is $s$ set equal to one.
Differentiating $G(s)$ a second time leads to
$$G''(s)=\sum_{x}s^{x-2}x(x-1)p(x),$$
so that, on putting $s=1$, we find
$$G''(1)=\sum_xx(x-1)p(x)=E[X(X-1)].$$
This is the second factorial moment of $W$, denoted by
$\mu_{[2]};$, from which the variance of $X$ can be readily
derived by using
\begin{equation}
\textrm{var}(X)=E[X(X-1)]+E(X)-[E(X)]^2=G''(1)+G'(1)-[G'(1)]^2.
\end{equation}
Continuing this process, if we differentiate $G(s)\ r$ times and
then set $s=1$, we find
$$G^{(r)}(1)=E[X(X-1)(X-2)\ldots(X-r+1)]=\mu'_{[r]},$$
which is the $r$th factorial moment of $X$. The $r$th moment of
$X$ about the origin, or about its mean, can be derived by writing
down a suitable function of that moment in terms of the first $r$
factorial moments, as we have just seen for the variance. (Note
that $\mu'_{[1]}=E(X)$.)
\item[Example 3]
Let us return to example 1, where we found that
$$G(s)=(1+s)^2(1+3s)/16$$
Differentiating with respect to $s$ yields
$$G'(s)=[3(1+s)^2+2(1+s)(1+3s)]/16=(1+s)(5+9s)/16.$$
Hence, from (2), the expected number of houses the builder will
sell is
$$E(X)=G'(1)=\frac{7}{4}.$$
Differentiating $G(s)$ a second time leads to
$$G''(s)=[9(1+s)+(5+9s)]/16=(7+9s)/;16.$$
Hence $G''(s)=2$ and, using (3), the variance of $X$ is found to
be
$$\textrm{var}(X)=G''(1)+G'(1)-[G'(1)]^2=2+\frac{7}{4}-
\left(\frac{7}{4}\right)^2=\frac{11}{16}.$$
\item[Linear functions of random variables]
Suppose $G_X(s)$ is the pgf of the random variable $X$ and we wish
to know the pgf $G_Y(s)$ of the linear function $Y=a+bX$. From
(1), we have
$$G_Y(s)=E(s^Y)=E(s^{a+bX}=s^aE[(s^b)^X]=s^aG_X(s^b)$$
so that $G_Y(s)$ can be found directly from $G_X(s)$.
\item[Example 4]
Let $X$ follow the distribution of Example 2 and let $Y=5+3X$. The
the pgf $G_Y(s)$ of $Y$ is given by
$$G_Y(s)=s^5G_X(s^3)=\frac{1}{4}s^5\left[1-\left(\frac{3s^3}{4}\right)
\right]^{-1}=\frac{1}{4}s^5\left[1+\frac{3}{4}s^3+
\left(\frac{3}{4}\right)^2s^6+\ldots\right].$$
Extracting the coefficient of $s^y$, we find
$$p_Y(y)=\frac{1}{4}\left(\frac{3}{4}\right)^{\frac{(y-5)}{3}},\
y=5,8,11,\ldots$$
\item[Binomial distribution revisited]
Suppose the random variable $Y$ follows a binomial $b)n\pi)$
distribution with probability function
$$p_Y(y)=\left(\begin{array}{c}n\\y\end{array}\right)\pi^y(1-\pi)^{n-y},\
y=0,1,\ldots,n.$$
The pgf of $Y$ is given by
$$G_Y(s)=\sum_{y=0}^Ns^yp_Y(y)=\sum_{y=0}^n\left(\begin{array}{c}n\\y
\end{array}\right)(\pi s)^y(1-\pi)^{n-y}=[(1-\pi)+\pi s]^n.$$
The mean and variance of $Y$ can be found by differentiating this
function. The first derivative with respect to $s$ is
$$G'_Y(s)=n\pi[(1-\pi)+\pi s]^{n-1},$$
so that
$$E(Y)=G'_Y(1)=n\pi.$$
The second derivative of $G_Y(s)$ is given by
$$G_Y''(s)=n(n-1)\pi^2[(1-\pi)+\pi s]^{n-2}.$$
From (3), we therefore find the variance of $Y$ to be
\begin{eqnarray*}
\textrm{var}(Y)&=&G''_Y(1)+G_Y'(1)-[G'_Y(1)]^2\\
&=&n(n-1)\pi^2+n\pi-(n\pi)^2\\ &=&n\pi(1-\pi).
\end{eqnarray*}
By differentiating $G_Y(s)$ twice more, we could go on to evaluate
the skewness and kurtosis of the distribution, although the
calculations become a little tedious.
\item[Poisson distribution]
Under certain conditions, the binomial distribution converges to a
different distribution as a limiting form. We have seen that the
pgf of a binomial variate is
$$G_Y(s)=[(1-\pi)+\pi s]^n.$$
Consider the limit as $n\to\infty$ and $\pi\to0$ in such a way as
the product $n\pi$ remains constant. In view of the restriction,
there is only a single limit at work here, and it is convenient,
in this analysis, to set $n\pi=\mu\ (n\pi$ is, after all, the mean
of the binomial distribution), to write $\pi=\frac{\mu}{n}$, and
to consider the limit of $G_Y(s)$ as $n\to\infty$. Then we have
\begin{equation}
\lim_{n\to\infty}G_Y(s)=\lim_{n\to\infty}
\left[1+\mu\frac{(s-1)}{n}\right]^n=e^{\mu(s-1)}.
\end{equation}
If $Y$ is a random variable with this pgf, its probability
function can be found by expanding (4) as a power series in $s$.
Since $e^{\mu s}=1+\frac{\mu s}{1!}+\frac{(\mu s)^2}{2!}+\ldots$,
this leads to
\begin{equation}
p_Y(y)=\frac{e^{-\mu}\mu^y}{y!},\ y=0,1,2,\ldots,
\end{equation}
which is the probability function of the \textit{Poisson
distribution}, named after the French mathematician Sim\'{e}on
Denis Poisson (1781-1840). Note that the range of $Y$ is
infinitely large as we have taken the limit as $n\to\infty$. And,
in view of the way the distribution has been derived, it will be
of no surprise to learn that it is particularly applicable in
situations where a large number $(n)$ of items or individuals are
\lq at risk ', each with a small probability $(\pi)$ of producing
an event, the variate $Y$ measuring the total number of events
over an interval of time or an area of space. If, for example, $Y$
is the number of particle emissions per unit time from a piece of
radioactive material, then $n$ is not only very large but also
unknown, and, because $\pi$ (also unknown) is extremely small, the
Poisson distribution fits data collected from such a source very
closely.
The mean and variance of $Y$ can be found by using (2) and (3).
Thus, since
$$G'_Y(s)=\mu e^{\mu(s-1)}\textrm{ and }G''_Y(s)=\mu^2
e^{\mu(s-1)},$$
we have that
$$E(Y)=\mu\textrm{ and var}(Y)=\mu^2+\mu-\mu^2=\mu.$$
These values can also be derived by taking the appropriate limits
of the mean and variance of the binomial distributions or, of
course, from the probability function (5), using the appropriate
summations. Being the mean value of $Y,\ \mu$ is a suitable symbol
to use for the parameter of the distribution.
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