The factorial moment-generating function (fmgf) for a univariate discrete distribution:

Compute an fmgf for a continuous univariate distribution:

The fmgf for a multivariate distribution:

Find the factorial moment-generating function (fmgf) for a discrete formula distribution:

Compute the fmgf for data distribution:

Find the fmgf for a censored distribution:

Compute the fmgf for parameter mixture distribution:

Find the fmgf for the slice distribution of a random process:

Find the fmgf for the sum of i.i.d. geometric variates:

Compare with the fmgf of NegativeBinomialDistribution:

Find the fmgf of the sum of a random number of i.i.d. geometric random variates, assuming follows PoissonDistribution:

Compare with the fmgf of PolyaAeppliDistribution:

Find the PDF of a non-negative integer random variate from its fmgf:

Use the probability generating function interpretation:

Show the probability mass function:

Verify normalization:

Construct a probability generating function for BernoulliDistribution:

Construct its Lagrange transformation, and use it as a new probability generating function:

Compare it with the probability generating function of a shifted GeometricDistribution:

Apply a Lagrange transformation to the probability generating function (pgf) of GeometricDistribution:

Reconstruct PDF:

The resulting distribution is known as Haight's distribution. It is only normalized to 1 for :

Show the probability mass function:

Find the distribution of the number of times a biased coin should be flipped until heads appear twice in a row. Let be the probability of heads. Event space is comprised of three types of events: tail (T), head then tail (HT), and two heads in a row (HH) with probabilities:

Find the fmgf of the random variate of interest, interpreting it as the total of the number of T events added to double the number of HT events plus 2:

Reconstruct PDF:

Compute mean:

Find variance using relation:

FactorialMomentGeneratingFunction is equivalent to Expectation of :

For non-negative discrete variates, the fmgf is the probability generating function (pgf):

The factorial moments can be extracted from the factorial moment-generating function:

Alternatively, use SeriesCoefficient:

For some distributions with long tails, factorial moments of only several low orders are defined:

Correspondingly, the factorial moment-generating function is not defined:

FactorialMomentGeneratingFunction is not always known in closed form: