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Statistics

  

Moment

  

compute moments

 

Calling Sequence

Parameters

Description

Computation

Data Set Options

Random Variable Options

Examples

References

Compatibility

Calling Sequence

Moment(A, n, ds_options)

Moment(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, origin, or weights; specify options for computing the moment of a data set

rv_options

-

(optional) equation(s) of the form option=value where option is one of numeric or origin; specifies options for computing the moment of a random variable

Description

• 

The Moment function computes the 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 Moment command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the Moment 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.

• 

origin=algebraic -- By default, the moment is computed about 0. If this option is present, the moment will be calculated about the specified point. If A is a Matrix, then you can specify several origins instead, one for each column of the matrix. This is accomplished by passing a list or Vector as the value of the origin option.

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 moment is computed symbolically. To compute the moment numerically, specify the numeric or numeric = true option.

• 

origin=algebraic -- By default, the moment is computed about 0. If this option is present, the moment will be calculated about the specified point.

Examples

withStatistics:

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

Moment'Β'3,5,3

112

(1)

Moment'Β'3,5,3,numeric

0.08333333333

(2)

Moment'Β'3,5,3,origin=12

196

(3)

Moment'Β'3,5,3,origin=12,numeric

0.01041666667

(4)

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

ASample'Β'3,5,105:

MomentA,3

0.0834247882855659

(5)

MomentA,3,origin=12

0.0104977275658779

(6)

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

XNormal5,2

X:=Normal5,2

(7)

BSampleX,106:

MomentX,3,StandardError[106]Moment,X,3,origin=0

185,15009465

(8)

MomentX,3,numeric,StandardError[106]Moment,X,3,origin=0,numeric

185.,0.1945764631

(9)

MomentB,3,StandardErrorMoment,B,3,origin=0

184.786341257454,0.194615225033025058

(10)

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

YRandomVariable'Β'5,2:

Moment1Y+2,3,numeric

0.05113729747

(11)

Verify this using simulation.

CSample1Y+2,105:

MomentC,3

0.0511321895166452

(12)

Compute the average 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:

MomentV,3,weights=W

HFloatundefined

(13)

MomentV,3,weights=W,ignore=true

3.02374062434479105

(14)

Consider the following Matrix data set.

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

M:=31130114694415271273683907884642878964844995128007

(15)

We compute the second moment of each of the columns.

MomentM,2

10.80000000000001.238437400000001061.257963093220001010

(16)

We compute the second moment of each column with origin 3.

MomentM,2,'origin=3'

0.6000000000000001.2319221061.257896492080001010

(17)

We compute the second moment of each column with three different origins.

MomentM,2,'origin=3,1000,100000'

0.60000000000000063637.40000000003.78950932200000108

(18)

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]

 


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