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.

FactorialMoment[r]

represents the r factorial moment.

Details

FactorialMoment handles both numerical and symbolic data.
For the list {x₁,x₂,…,x_n}, the r factorial moment is given by .
FactorialMoment[{{x₁,y₁,…},…,{x_n,y_n,…}},{r_x,r_y,…}] gives .
FactorialMoment works with SparseArray objects.
For a distribution dist, the r factorial moment is given by Expectation[FactorialPower[x,r],xdist].
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

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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)

TraditionalForm formatting for formal moments:

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 $mu^__r=sum_(k=1)^rTemplateBox[{r, k}, StirlingS1]mu_k$ :

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:

Neat Examples (1)

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

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FactorialMoment

Details

Examples

Basic Examples (2)