FactorialMoment [data,r]
gives the order r factorial moment of data.
FactorialMoment [data,{r1,…,rm}]
gives the order {r1,…,rm} multivariate factorial moment of data.
FactorialMoment [dist,…]
gives the factorial moment of the distribution dist.
FactorialMoment [r]
represents the order r formal factorial moment.
FactorialMoment
FactorialMoment [data,r]
gives the order r factorial moment of data.
FactorialMoment [data,{r1,…,rm}]
gives the order {r1,…,rm} multivariate factorial moment of data.
FactorialMoment [dist,…]
gives the factorial moment of the distribution dist.
FactorialMoment [r]
represents the order r formal factorial moment.
Details
- Factorial moments are defined using FactorialPower [x,r] given by .
- For scalar order r and data being an array :
-
x in TemplateBox[{Vectors, paclet:ref/Vectors}, RefLink, BaseStyle -> {3ColumnTableMod}][n] sum of r^(th) factorial powers »x in TemplateBox[{Matrices, paclet:ref/Matrices}, RefLink, BaseStyle -> {3ColumnTableMod}][{n,m}] columnwise sum of r^(th) factorial powers »x in TemplateBox[{Arrays, paclet:ref/Arrays}, RefLink, BaseStyle -> {3ColumnTableMod}][{n_(1),...,n_(k)}] columnwise sum of r^(th) factorial powers »
- FactorialMoment [x,r] is equivalent to ArrayReduce [FactorialMoment[#,r]&,x,1].
- For vector order {r1,…,rm} and data being array :
-
x in TemplateBox[{Matrices, paclet:ref/Matrices}, RefLink, BaseStyle -> {3ColumnTableMod}][{n,m}] sum the rj^(th) factorial power in the j^(th) columnx in TemplateBox[{Arrays, paclet:ref/Arrays}, RefLink, BaseStyle -> {3ColumnTableMod}][{n_(1),...,n_(k)}] sum the rj^(th) factorial power in the j^(th) column »
- FactorialMoment [x,{r1,…,rm}] is equivalent to ArrayReduce [FactorialMoment[#,{r1,…,rm}]&,x,{{1},{2}}].
- FactorialMoment handles both numerical and symbolic data.
- The data can have the following additional forms and interpretations:
-
Association the values (the keys are ignored) »WeightedData weighted mean, based on the underlying EmpiricalDistribution »EventData based on the underlying SurvivalDistribution »
- For a distribution dist, the r^(th) factorial moment is given by Expectation [x(r),xdist]. »
- For a multivariate distribution dist, the {r1,…,rm}^(th) factorial moment is given by Expectation [x1(r1)⋯xm(rm),{x1,…,xm}dist]. »
- For a random process proc, the factorial moment function can be computed for slice distribution at time t, SliceDistribution [proc,t], as mu^__r[t]=FactorialMoment[SliceDistribution [proc,t],r]. »
- FactorialMoment [r] can be used in such functions as MomentConvert and MomentEvaluate , etc. »
Examples
open all close allBasic 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 (20)
Basic Uses (5)
Exact input yields exact output:
Approximate input yields approximate output:
Find factorial moments of WeightedData :
Find a factorial moment of EventData :
Find a factorial moment of TimeSeries :
Factorial moment depends only on the values:
Array Data (4)
For a matrix, FactorialMoment gives columnwise moments:
For an array, FactorialMoment gives columnwise moments at the first level:
Multivariate FactorialMoment for an array:
Works with large arrays:
When the input is an Association , FactorialMoment works on its values:
SparseArray data can be used just like dense arrays:
Image and Audio Data (2)
Channelwise factorial moment of an RGB image:
Factorial moment intensity value of a grayscale image:
On audio objects, FactorialMoment works channelwise:
Distribution and Process Moments (5)
Scalar factorial moment for univariate distributions:
Scalar factorial moment for multivariate distributions:
Joint factorial moment for 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 factorial moment function together with all the simulations:
Formal Moments (4)
TraditionalForm formatting for formal moments:
Convert combinations of formal moments to an expression involving FactorialMoment :
Evaluate an expression involving formal moments TemplateBox[{2}, FactorialMoment]+TemplateBox[{3}, FactorialMoment] 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 factorial 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 (5)
Factorial moment is equivalent to an expectation of FactorialPower :
First factorial moment is equivalent to Mean :
FactorialMoment can be computed from Moment through mu^__r=sum_(k=1)^rTemplateBox[{r, k}, StirlingS1]mu_k :
MomentConvert produces the same result:
Moment can be computed from FactorialMoment through mu_r=sum_(k=0)^rmu^__k TemplateBox[{r, k}, StirlingS2]:
MomentConvert produces the same result:
The multivariate factorial moment of an array of depth has depth :
Neat Examples (1)
The distribution of FactorialMoment estimates for 30, 100, and 300 samples:
Text
Wolfram Research (2010), FactorialMoment, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorialMoment.html (updated 2024).
CMS
Wolfram Language. 2010. "FactorialMoment." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. 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_2025_factorialmoment, author="Wolfram Research", title="{FactorialMoment}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/FactorialMoment.html}", note=[Accessed: 04-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_factorialmoment, organization={Wolfram Research}, title={FactorialMoment}, year={2024}, url={https://reference.wolfram.com/language/ref/FactorialMoment.html}, note=[Accessed: 04-December-2025]}