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Statistics[CentralMoment] - compute the central moments

Calling Sequence

CentralMoment(A, n, ds_options)

CentralMoment(M, n, ds_options)

CentralMoment(X, n, rv_options)

Parameters

A

-

data sample

M

-

Matrix data set

X

-

algebraic; random variable or distribution

n

-

algebraic; order

ds_options

-

(optional) equation(s) of the form option=value where option is one of ignore, or weights; specify options for computing the central moment of a data set

rv_options

-

(optional) equation of the form numeric=value; specifies options for computing the central moment of a random variable

Description

• 

The CentralMoment function computes the central moment of order n of the specified random variable or data set.

• 

The first parameter can be a data set (e.g., a Vector), a Matrix data set, a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

• 

The second parameter, n, can be any Maple expression.

Computation

• 

All computations involving data are performed in floating-point; therefore, all data provided must have type realcons and all returned solutions are floating-point, even if the problem is specified with exact values.

• 

By default, all computations involving random variables are performed symbolically (see option numeric below).

• 

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

Data Set Options

  

The ds_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[DescriptiveStatistics] help page.

• 

ignore=truefalse -- This option controls how missing data is handled by the CentralMoment command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the CentralMoment command will return undefined. If ignore=true all missing items in A will be ignored. The default value is false.

• 

weights=Vector -- Data weights. The number of elements in the weights array must be equal to the number of elements in the original data sample. By default all elements in A are assigned weight 1.

Random Variable Options

  

The rv_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

• 

numeric=truefalse -- By default, the central moment is computed symbolically. To compute the central moment numerically, specify the numeric or numeric = true option.

Examples

withStatistics:

Compute the third central moment of the beta distribution with parameters 3 and 5.

CentralMoment'Β'3,5,3

1768

(1)

CentralMoment'Β'3,5,3,numeric

0.001302083333

(2)

Generate a random sample of size 100000 drawn from the above distribution and compute the third central moment.

A:=Sample'Β'3,5,105:

CentralMomentA,3

0.00134333467162571

(3)

Compute the standard error of the fourth central moment for the normal distribution with parameters 5 and 2.

X:=RandomVariableNormal5,2:

B:=SampleX,106:

CentralMomentB,4,StandardErrorCentralMoment,B,4

48.2355377064000,0.163721302392168894

(4)

Create a beta-distributed random variable Y and compute the third central moment of 1Y+2.

Y:=RandomVariable'Β'5,2:

CentralMoment1Y+2,3,numeric

0.00001053304160

(5)

Verify this using simulation.

C:=Sample1Y+2,105:

CentralMomentC,3

0.0000106293550474148

(6)

Compute the average central moment of a weighted data set.

V:=seqi,i=57..77,undefined:

W:=2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5:

CentralMomentV,4,weights=W

HFloatundefined

(7)

CentralMomentV,4,weights=W,ignore=true

137.689183427122

(8)

Consider the following Matrix data set.

M:=Matrix3,1130,114694,4,1527,127368,3,907,88464,2,878,96484,4,995,128007

M:=31130114694415271273683907884642878964844995128007

(9)

We compute the third central moment of each column.

CentralMomentM,3

0.1440000000000001.383749956800001071.032522936468431012

(10)

See Also

Statistics, Statistics[Computation], Statistics[DescriptiveStatistics], Statistics[Distributions], Statistics[ExpectedValue], Statistics[RandomVariables], Statistics[StandardError]

References

  

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998.  Vol. 1: Distribution Theory.


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