probability-generating function

The latter is therefore called the probability-generating function of the former.

Probability-generating functions obey all the rules of power series with non-negative coefficients.

Related concepts include the moment-generating function and the probability-generating function.

Probability-generating functions are particularly useful for dealing with functions of independent random variables.

The probability-generating function is an example of a generating function of a sequence: see also formal power series.

This report, which invokes probability-generating functions, is also an early entry in the extensive literature on statistics of branching and multiplicative processes.

Note that this is the n-fold product of the probability-generating function of a Bernoulli random variable with parameter p.

So the radius of convergence of any probability-generating function must be at least 1, by Abel's theorem for power series with non-negative coefficients.

Probability-generating function / (1:D)

If a is the probability mass function of a discrete random variable, then its ordinary generating function is called a probability-generating function.

Suppose again that N is also an independent, discrete random variable taking values on the non-negative integers, with probability-generating function G and probability density .

One context in which factorial moments occur naturally is at an initial stage in the use of probability-generating functions to derive the moments of discrete distributions.

This is the partial sum of the infinite series giving the exponential function at z 1, which in turn is the probability-generating function of the Poisson distribution with parameter 1.

G(z) is called the generating function of the sequence a. Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negative sequences, such as probability-generating functions.