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Statistics

  

StandardizedMoment

  

compute standardized moments

 

Calling Sequence

Parameters

Description

Computation

Data Set Options

Random Variable Options

Examples

References

Compatibility

Calling Sequence

StandardizedMoment(A, n, ds_options)

StandardizedMoment(X, n, rv_options)

Parameters

A

-

data set or 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 standardized moment of a data set

rv_options

-

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

Description

• 

The StandardizedMoment function computes the standardized 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 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 StandardizedMoment command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the StandardizedMoment 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 standardized moment is computed symbolically. To compute the standardized moment numerically, specify the numeric or numeric = true option.

Examples

withStatistics:

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

StandardizedMoment'Β'3,5,3

21525

(1)

StandardizedMoment'Β'3,5,3,numeric

0.3098386677

(2)

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

ASample'Β'3,5,105:

StandardizedMomentA,3

0.315910062621570

(3)

Create a beta-distributed random variable Y and compute the third standardized moment of 1/(Y+2).

YRandomVariable'Β'5,2:

StandardizedMoment1Y+2,3,numeric

0.8947305775

(4)

Verify this using simulation.

CSample1Y+2,105:

StandardizedMomentC,3

0.896000235993627

(5)

Compute the average standardized moment of a weighted data set.

Vseqi,i=57..77,undefined:

W2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5:

StandardizedMomentV,4,weights=W

Floatundefined

(6)

StandardizedMomentV,4,weights=W,ignore=true

2.48823188102473

(7)

Consider the following Matrix data set.

MMatrix3,1130,114694,4,1527,127368,3,907,88464,2,878,96484,4,995,128007

M31130114694415271273683907884642878964844995128007

(8)

We compute fourth standardized moment of each of the columns.

StandardizedMomentM,4

1.182040816326531.649175573998300.881611283129665

(9)

References

  

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

Compatibility

• 

The A parameter was updated in Maple 16.

See Also

Statistics

Statistics[Computation]

Statistics[DescriptiveStatistics]

Statistics[Distributions]

Statistics[ExpectedValue]

Statistics[RandomVariables]

Statistics[StandardError]