Editing Probability-generating function (section)

==Examples==

* The probability generating function of an almost surely [[degenerate distribution|constant random variable]], i.e. one with <math>Pr(X=c) = 1</math> and <math>Pr(X\neq c) = 0</math> is <math display="block">G(z) = z^c. </math>
* The probability generating function of a [[binomial distribution|binomial random variable]], the number of successes in <math>n</math> trials, with probability <math>p</math> of success in each trial, is <math display="block">G(z) = \left[(1-p) + pz\right]^n. </math> '''Note''': it is the <math>n</math>-fold product of the probability generating function of a [[Bernoulli distribution|Bernoulli random variable]] with parameter <math>p</math>. {{pb}}  So the probability generating function of a [[fair coin]], is <math display="block">G(z) = 1/2 + z/2. </math>
* The probability generating function of a [[negative binomial distribution|negative binomial random variable]] on <math>\{0,1,2 \cdots\}</math>, the number of failures until the <math>r^{th}</math> success with probability of success in each trial <math>p</math>, is <math display="block">G(z) = \left(\frac{p}{1 - (1-p)z}\right)^r,</math> which converges for <math>|z| < \frac{1}{1-p}</math>. {{pb}} '''Note''' that this is the <math>r</math>-fold product of the probability generating function of a [[geometric distribution|geometric random variable]] with parameter <math>1-p</math> on <math>\{0,1,2,\cdots\}</math>.
* The probability generating function of a [[Poisson distribution|Poisson random variable]] with rate parameter <math>\lambda</math> is <math display="block">G(z) = e^{\lambda(z - 1)}.</math>
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==Joint probability generating functions==

The concept of the probability generating function for single random variables can be extended to the joint probability generating function of two or more random variables.

Suppose that ''X'' and ''Y'' are both discrete random variables (not necessarily independent or identically distributed), again taking values on some subset of the non-negative integers. -->