Factorial moment

The

r

E [(N)_{r}]

, where

(N)_{r} = N (N - 1) \dots (N - r + 1)

Factorial-moment inversion ( $Pr (X = k) = \frac{1}{k !} \sum_{r \geq 0} \frac{( - 1 ) ^{r} E ( X ) _{k + r}}{r !}$ )

For a nonnegative integer-valued random variable, the displayed inversion is valid when

E (X)_{s} s^{k} / s! \to 0

E (X)_{k + m + 1} / [k! (m + 1)!]

, which tends to zero under this hypothesis. No exchange of an infinite signed sum with an expectation is needed.

N

is Poisson with mean

λ

, then

E [(N)_{r}] = λ^{r}

; in particular,

E N = λ

and

E [N (N - 1)] = λ^{2}

If nonnegative integer-valued random variables

X_{n}

have all fixed factorial moments converging to

λ^{k}

λ

. The first-moment bound gives uniform tightness, and bounds on higher moments give uniform integrability of each fixed lower moment. Every subsequential limit therefore has the indicated factorial moments. For such a limit

X

and

t \geq 0

, the nonnegative binomial expansion and the monotone convergence theorem give

E (1 + t)^{X} = \sum_{k \geq 0} t^{k} λ^{k} / k! = e^{λ t}

. This identifies its moment-generating function as that of the Poisson distribution, uniquely determining the limit.

There are no discussions about this article yet.

The term "factorial moment" refers to a specific type of moment used in probability theory and statistics. Factorial moments are particularly useful when dealing with discrete random variables, especially in the context of counting and combinatorial problems. For a discrete random variable $ X $ taking non-negative integer values, the $ n $-th factorial moment is defined as: \[ E[X^{(n)}] = E\left[\frac{X!}{(X-n)!