# FactorialMoment

FactorialMoment[list,r]

gives the r moment of the elements in the list.

FactorialMoment[dist,r]

gives the r moment of the distribution dist.

represents the r factorial moment.

# Examples

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## Basic Examples(2)

Compute factorial moment from data:

Use symbolic data:

Compute the second factorial moment of a discrete univariate distribution:

The factorial moment for a multivariate distribution:

## Scope(18)

### Data Moments(9)

Exact input yields exact output:

Approximate input yields approximate output:

FactorialMoment for a matrix gives column-wise means:

FactorialMoment for a tensor gives column-wise means at the first level:

Work with large arrays:

SparseArray data can be used just like dense arrays:

Find factorial moments of WeightedData:

Find a factorial moment of EventData:

Find a factorial moment of TimeSeries:

The moment depends only on the values:

Find a factorial moment for data involving quantities:

### Distribution and Process Moments(5)

Find the factorial moments for univariate distributions:

Multivariate distributions:

Compute a factorial moment for a symbolic order r:

A factorial moment may only evaluate for specific orders:

A factorial moment may only evaluate numerically:

Factorial moments for derived distributions:

Data distribution:

Factorial moment function for a random process:

Find a factorial moment of TemporalData at some time t=0.5:

Find the corresponding moment function together with all the simulations:

### Formal Moments(4)

Convert combinations of formal moments to an expression involving FactorialMoment:

Evaluate an expression involving formal moments for a distribution:

Evaluate for data:

Find a sample estimator for an expression involving FactorialMoment:

Evaluate the resulting estimator for data:

## Applications(4)

Estimate parameters of a distribution using the method of moments:

Compare data and the estimated parametric distribution:

Reconstruct probability mass function from the sequence of factorial moments:

Find the factorial moment-generating function (fmgf):

Use equivalence of the fmgf and the probability generating function:

Verify that factorial moments of the found distribution match the originals:

Compute a moving factorial moment for some data:

Use the window of length .1:

Compute factorial moments for slices of a collection of paths of a random process:

Choose a few slice times:

Plot factorial moments over these paths:

## Properties & Relations(4)

Factorial moment is equivalent to an expectation of FactorialPower:

First factorial moment is equivalent to Mean:

FactorialMoment can be computed from Moment through :

MomentConvert produces the same result:

Moment can be computed from FactorialMoment through :

MomentConvert produces the same result:

## Neat Examples(1)

The distribution of FactorialMoment estimates for 30, 100, and 300 samples:

Wolfram Research (2010), FactorialMoment, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorialMoment.html.

#### Text

Wolfram Research (2010), FactorialMoment, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorialMoment.html.

#### CMS

Wolfram Language. 2010. "FactorialMoment." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FactorialMoment.html.

#### APA

Wolfram Language. (2010). FactorialMoment. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FactorialMoment.html

#### BibTeX

@misc{reference.wolfram_2023_factorialmoment, author="Wolfram Research", title="{FactorialMoment}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/FactorialMoment.html}", note=[Accessed: 30-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_factorialmoment, organization={Wolfram Research}, title={FactorialMoment}, year={2010}, url={https://reference.wolfram.com/language/ref/FactorialMoment.html}, note=[Accessed: 30-September-2023 ]}