Factorial moment generating function

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

$M_X(t)=\operatorname{E}\bigl[t^{X}\bigr]$

for all complex numbers t for which this expected value exists. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable In Mathematics, the real numbers may be described informally in several different ways A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This is the case at least for all t on the unit circle $| t | = 1$ , see characteristic function. In Mathematics, a unit circle is In Probability theory, the characteristic function of any Random variable completely defines its Probability distribution. If X is a discrete random variable taking values only in the set {0,1, . . . } of non-negative integers, then $M X$ is also called probability-generating function of X and $M X (t)$ is well-defined at least for all t on the closed unit disk $|t|\le1$ . The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Probability theory, the probability-generating function of a Discrete random variable is a Power series representation (the Generating function In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematics, the open unit disk around P (where P is a given point in the plane) is the set of points whose distance from P is

The factorial moment generating function generates the factorial moments of the probability distribution. In Probability theory, the n th factorial moment of a Probability distribution, also called the n th factorial moment of any Random variable In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable Provided $M X$ exists in a neighbourhood of t = 1, the nth factorial moment is given by

$\operatorname{E}[(X)_n]=M_X^{(n)}(1)=\left.\frac{\mathrm{d}^n}{\mathrm{d}t^n}\right|_{t=1} M_X(t),$

where the Pochhammer symbol (x)_n is the falling factorial

$(x)_n = x(x-1)(x-2)\cdots(x-n+1).\,$

(Confusingly, some mathematicians, especially in the field of special functions, use the same notation to represent the rising factorial. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In Mathematics, the Pochhammer symbol (x_{n}\ introduced by Leo August Pochhammer, represents either the rising or the falling In Mathematics, the Pochhammer symbol (x_{n}\ introduced by Leo August Pochhammer, represents either the rising or the falling Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the Mathematical analysis In Mathematics, the Pochhammer symbol (x_{n}\ introduced by Leo August Pochhammer, represents either the rising or the falling )

Example

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

$M_X(t) =\sum_{k=0}^\infty t^k\underbrace{\operatorname{P}(X=k)}_{=\,\lambda^ke^{-\lambda}/k!} =e^{-\lambda}\sum_{k=0}^\infty \frac{(t\lambda)^k}{k!} = e^{\lambda(t-1)},\qquad t\in\mathbb{C},$

(use the definition of the exponential function) and thus we have

$\operatorname{E}[(X)_n]=\lambda^n.$

Example

See also