Factorial moment generating function explained

In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable X is defined as

M ^{X} _{X(t) = \operatorname{E} l [t} r ] for all complex numbers t for which this expected value exists. This is the case at least for all t on the unit circle |t| =1 , see characteristic function. If X is a discrete random variable taking values only in the set of non-negative integers, then M _{X} is also called probability-generating function (PGF) of X and M _{X(t)} is well-defined at least for all t on the closed unit disk |t|\le 1 .

The factorial moment generating function generates the factorial moments of the probability distribution.Provided

M _{X} exists in a neighbourhood of t = 1, the nth factorial moment is given by ^[1] \operatorname{E}[(X) _{n]=M} ( 1 ) = \left. ^{(n)} _{X} (d ^{n})/(d t ^{n}) \right| _{t=1} M _{X(t),} where the Pochhammer symbol (x)_n is the falling factorial(x) _{n} = x(x -1 )(x -2 ) … (x - n +1 ). (Many mathematicians, especially in the field of special functions, use the same notation to represent the rising factorial.)

Examples

Poisson distribution

Suppose X has a Poisson distribution with expected value λ, then its factorial moment generating function is

M _{X(t) =\sum} ^{inf ty} ^{k =0} t ^{k\underbrace{\operatorname{P}(X = k)}} _{=λ ^{ke} ^{-λ} /k!} = e ^{-λ} \sum ^{inf ty} _{k =0} ((t λ ) ^{k})/(k!) = e ^{λ(t-1)} , t\in C , (use the definition of the exponential function) and thus we have\operatorname{E}[(X) ^{n.} _{n] =λ}

See also

• Moment (mathematics)
• Moment-generating function
• Cumulant-generating function

Notes and References

1. Web site: Néri . Breno de Andrade Pinheiro . 2005-05-23 . Generating Functions . https://web.archive.org/web/20120331042031/https://files.nyu.edu/bpn207/public/Teaching/2005/Stat/Generating_Functions.pdf . 2012-03-31 . nyu.edu.