Factorial moment

Template:Short description In probability theory, the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments are useful for studying non-negative integer-valued random variables,^[1] and arise in the use of probability-generating functions to derive the moments of discrete random variables.

Factorial moments serve as analytic tools in the mathematical field of combinatorics, which is the study of discrete mathematical structures.^[2]

Definition

For a natural number Template:Math, the Template:Math-th factorial moment of a probability distribution on the real or complex numbers, or, in other words, a random variable Template:Math with that probability distribution, is^[3]

E [(X)_{r}] = E [X (X - 1) (X - 2) \dots (X - r + 1)],

where the Template:Math is the expectation (operator) and

(x)_{r} : = \underset{r factors}{\underset{⏟}{x (x - 1) (x - 2) \dots (x - r + 1)}} \equiv \frac{x!}{(x - r)!}

is the falling factorial, which gives rise to the name, although the notation Template:Math varies depending on the mathematical field.Template:Efn Of course, the definition requires that the expectation is meaningful, which is the case if Template:Math or Template:Math.

If Template:Math is the number of successes in Template:Math trials, and Template:Math is the probability that any Template:Math of the Template:Math trials are all successes, then^[4]

E [(X)_{r}] = n (n - 1) (n - 2) \dots (n - r + 1) p_{r}

Examples

Poisson distribution

If a random variable Template:Math has a Poisson distribution with parameter λ, then the factorial moments of Template:Math are

E [(X)_{r}] = λ^{r},

which are simple in form compared to its moments, which involve Stirling numbers of the second kind.

Binomial distribution

If a random variable Template:Math has a binomial distribution with success probability Template:Math Template:Closed-closed and number of trials Template:Math, then the factorial moments of Template:Math are^[5]

E [(X)_{r}] = (\binom{n}{r}) p^{r} r! = (n)_{r} p^{r},

where by convention, $(\binom{n}{r})$ and $(n)_{r}$ are understood to be zero if r > n.

Hypergeometric distribution

If a random variable Template:Math has a hypergeometric distribution with population size Template:Math, number of success states Template:Math} in the population, and draws Template:Math}, then the factorial moments of Template:Math are ^[5]

E [(X)_{r}] = \frac{(\binom{K}{r}) (\binom{n}{r}) r!}{(\binom{N}{r})} = \frac{(K)_{r} (n)_{r}}{(N)_{r}} .

Beta-binomial distribution

If a random variable Template:Math has a beta-binomial distribution with parameters Template:Math, Template:Math, and number of trials Template:Math, then the factorial moments of Template:Math are

E [(X)_{r}] = (\binom{n}{r}) \frac{B (α + r, β) r!}{B (α, β)} = (n)_{r} \frac{B (α + r, β)}{B (α, β)}

Calculation of moments

The rth raw moment of a random variable X can be expressed in terms of its factorial moments by the formula

E [X^{r}] = \sum_{j = 1}^{r} {\binom{r}{j}} E [(X)_{j}],

where the curly braces denote Stirling numbers of the second kind.

Notes

Template:Notelist

References

Template:Reflist

↑ D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol. I. Probability and its Applications (New York). Springer, New York, second edition, 2003
↑ Template:Cite book
↑ Template:Cite book
↑ P.V.Krishna Iyer. "A Theorem on Factorial Moments and its Applications". Annals of Mathematical Statistics Vol. 29 (1958). Pages 254-261.
↑ ^5.0 ^5.1 Template:Cite journal

[daleyPPI2003-1] D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol. I. Probability and its Applications (New York). Springer, New York, second edition, 2003

[2] Template:Cite book

[3] Template:Cite book

[4] P.V.Krishna Iyer. "A Theorem on Factorial Moments and its Applications". Annals of Mathematical Statistics Vol. 29 (1958). Pages 254-261.

[potts1953note-5] 5.0 ^5.1 Template:Cite journal

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Factorial moment

Contents

Definition

Examples

Poisson distribution

Binomial distribution

Hypergeometric distribution

Beta-binomial distribution

Calculation of moments

See also

Notes

References

Navigation menu

Factorial moment

Definition

Examples

Poisson distribution

Binomial distribution

Hypergeometric distribution

Beta-binomial distribution

Calculation of moments

See also

Notes

References

Navigation menu

Search