BUILT-IN MATHEMATICA SYMBOL

CentralMoment

gives the r

central moment of the elements in list with respect to their mean.

CentralMoment

gives the r

central moment of the symbolic distribution dist.

CentralMoment[r]
represents the r

formal central moment.

MORE INFORMATION

CentralMoment handles both numerical and symbolic data.

For the list , the central moment is given by , where is the mean of the list.

CentralMoment gives .

CentralMoment works with SparseArray objects.

For a symbolic distribution dist, the rcentral moment is given by Expectation[(x-Mean[dist])^r, xdist].

For a multivariate symbolic distribution dist, the central moment is given by Expectation and {₁, ₂, ...}==Mean[dist].

CentralMoment[r] can be used in functions such as MomentConvert and MomentEvaluate, etc.

EXAMPLES

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Basic Examples (5)

Compute central moments from data:

Use symbolic data:

Compute the second central moment of a continuous univariate distribution:

The central moment

of a discrete univariate distribution:

The central moment

of a multivariate distribution:

Find the relation of a formal central moment to cumulants:

Evaluate for a particular distribution:

Compute central moments from data:

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Use symbolic data:

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Compute the second central moment of a continuous univariate distribution:

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The central moment

of a discrete univariate distribution:

In[1]:=

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The central moment

of a multivariate distribution:

In[1]:=

Out[1]=

Find the relation of a formal central moment to cumulants:

In[1]:=

Out[1]=

Evaluate for a particular distribution:

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Scope (10)

Compute central moment

for a univariate distribution:

Compute a central moment of a specific order:

Evaluate a central moment of specific order numerically:

Compute central moments of a multivariate distribution:

Compute a central moment of a truncated distribution:

Find central moment

of a formula-based distribution:

Evaluate central moment

of a function of random variables:

Compute a central moment of distributions derived from data:

Compute central moment

for a set of 5 independent identically distributed samples:

TraditionalForm formatting:

Generalizations & Extensions (1)

Compute results for a SparseArray:

Compute a multivariate central moment:

Applications (6)

Estimate parameters of a distribution using the method of moments:

Compare data and the estimated parametric distribution:

Find a normal approximation to GammaDistribution using the method of moments:

Show how

and

depend on

and

:

Compare an original and an approximated distribution:

Construct a sample estimator of the second central moment:

Find its sample distribution expectation, assuming sample size

:

Find sample distribution variance of the estimator:

Variance of the estimator for uniformly distributed sample:

The law of large numbers states that a sample moment approaches population moment as sample size increases. Use Histogram to show the probability distribution of a second sample central moment of uniform random variates for different sample sizes:

Edgeworth expansion for near-normal data correcting for third and fourth central moments:

Function computing sample Jarque-Bera statistics []:

Accumulate statistics on samples of normal random variates:

Compare the statistics histogram with an asymptotic distribution:

Properties & Relations (10)

The first central moment is 0:

Central moments are translation invariant:

The second central moment is a scaled Variance:

Sqrt of the second central moment is RootMeanSquare of deviations from the Mean:

Skewness is a ratio of powers of third and second central moments:

Kurtosis is a ratio of powers of fourth and second central moments:

CentralMoment is equivalent to an Expectation of a power of a random variable around its mean:

CentralMoment of order

is equivalent to

when both exist:

Use CentralMoment directly:

Find the central moment generating function by using GeneratingFunction:

Compare with direct evaluation of CentralMomentGeneratingFunction:

CentralMoment can be expressed in terms of Moment, Cumulant, or FactorialMoment:

Possible Issues (3)

CentralMoment

was modified to compute a multivariate central moment:

Column-wise marginal moments can be computed as follows:

Alternatively, evaluate dimension-wise:

Moments of higher order are undefined for a heavy-tailed distribution:

Compute central moments on 5 independent samples of the distribution:

Sample central moments of higher order exhibit wild fluctuations:

Sample estimators of central moments are biased:

Find sampling population expectation assuming a sample of size

:

The estimator is asymptotically unbiased:

Construct an unbiased estimator:

The expected value of the estimator is the central moment for all sample sizes: