Editing Probability-generating function (section)

===Functions of independent random variables===

Probability generating functions are particularly useful for dealing with functions of [[statistical independence|independent]] random variables. For example:

* If <math>X_i, i=1,2,\cdots,N</math> is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and
<math display="block">S_N = \sum_{i=1}^N a_i X_i,</math> where the <math>a_i</math> are constant natural numbers, then the probability generating function is given by
<math display="block">G_{S_N}(z) = \operatorname{E}(z^{S_N}) = \operatorname{E} \left( z^{\sum_{i=1}^N a_i X_i,} \right) = G_{X_1}( z^{a_1})G_{X_2}(z^{a_2})\cdots G_{X_N}(z^{a_N}).</math>
* In particular, if <math>X</math> and <math>Y</math> are independent random variables:
<math display="block">G_{X+Y}(z) = G_X(z) \cdot G_Y(z)</math> and
<math display="block">G_{X-Y}(z) = G_X(z) \cdot G_Y(1/z).</math>
* In the above, the number <math>N</math> of independent random variables in the sequence is fixed. Assume <math>N</math> is discrete random variable taking values on the non-negative integers, which is independent of the <math>X_i</math>, and consider the probability generating function <math>G_N</math>.  If the <math>X_i</math> are not only independent but also identically distributed with common probability generating function <math>G_X = G_{X_i}</math>, then
<math display="block">G_{S_N}(z) = G_N(G_X(z)).</math> This can be seen, using the [[law of total expectation]], as follows:
<math display="block">
\begin{align}
G_{S_N}(z) & = \operatorname{E}(z^{S_N}) = \operatorname{E}(z^{\sum_{i=1}^N X_i}) \\[4pt]
& = \operatorname{E}\big(\operatorname{E}(z^{\sum_{i=1}^N X_i} \mid N) \big) = \operatorname{E}\big( (G_X(z))^N\big) =G_N(G_X(z)).
\end{align}
</math>
This last fact is useful in the study of [[Galton&ndash;Watson process]]es and [[compound Poisson process]]es.
* When the <math>X_i</math> are not supposed identically distributed (but still independent and independent of <math>N</math>), we have
<math display="block">G_{S_N}(z) = \sum_{n \ge 1} f_n \prod_{i=1}^n G_{X_i}(z),</math> where <math>f_n = \Pr(N=n).</math> For identically distributed <math>X_i</math>s, this simplifies to the identity stated before, but the general case is sometimes useful to obtain a decomposition of <math>S_N</math> by means of generating functions.