Probability-generating function

In probability theory, the probability generating function of a - valued random variable is defined by

Is there.

Properties

• The probability generating function determines the distribution of unambiguous:

• Are the random variables independent - valued random variables, then:

• For a - valued random variables with finite th moment is

Examples

• For a Bernoulli random variable with and apply.
• Since a binomial random variable can be written with the parameters and the sum of independent random variables Bernoulli distributed, follows.
• Is uniformly distributed on, then applies

• When is geometrically distributed with parameter arises.
• It follows that a negative binomial random variable has the probability generating function.
• For Poisson distributed with parameter