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.
Details

- FactorialMoment handles both numerical and symbolic data.
- For the list {x1,x2,…,xn}, the r
factorial moment is given by
.
- FactorialMoment[{{x1,y1,…},…,{xn,yn,…}},{rx,ry,…}] gives
.
- FactorialMoment works with SparseArray objects.
- For a distribution dist, the r
factorial moment is given by Expectation[FactorialPower[x,r],xdist].
- For a multivariate distribution dist, the {r1,r2,…}
factorial moment is given by Expectation[FactorialPower[x1,r1]FactorialPower[x2,r2]…,{x1,x2,…}dist].
- FactorialMoment[r] can be used in such functions as MomentConvert and MomentEvaluate, etc.
Examples
open allclose allBasic Examples (2)
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:
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:
Distribution and Process Moments (5)
Find the factorial moments for univariate 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:
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)
TraditionalForm formatting for formal moments:
Convert combinations of formal moments to an expression involving FactorialMoment:
Evaluate an expression involving formal moments for a distribution:
Find a sample estimator for an expression involving FactorialMoment:
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:
Compute factorial moments for slices of a collection of paths of a random process:
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:
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