# 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].
• can be used in such functions as MomentConvert and MomentEvaluate, etc.

# 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: