This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# FactorialMomentGeneratingFunction

 FactorialMomentGeneratingFunction gives the factorial moment generating function for the symbolic distribution dist as a function of the variable t. FactorialMomentGeneratingFunctiongives the factorial moment generating function for the multivariate symbolic distribution dist as a function of the variables , , ....
• The i factorial moment can be extracted from a factorial moment generating function fmgf through SeriesCoefficient[fmgf, {t, 1, i}]i!.
• The probability for a discrete random variable to assume the value i can be extracted from a factorial moment generating function expr through SeriesCoefficient.
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:
The factorial moment generating function (fmgf) for a univariate discrete distribution:
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Compute an fmgf for a continuous univariate distribution:
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The fmgf for a multivariate distribution:
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 Scope   (4)
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:
 Applications   (6)
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:
For non-negative discrete variates, the fmgf is the probability generating function (pgf):
The factorial moment generating function is the exponential generating function for factorial moments:
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:
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