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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 - M - 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).

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

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$

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

 > $\mathrm{CentralMoment}\left('\mathrm{Β}'\left(3,5\right),3\right)$
 $\frac{{1}}{{768}}$ (1)
 > $\mathrm{CentralMoment}\left('\mathrm{Β}'\left(3,5\right),3,\mathrm{numeric}\right)$
 ${0.001302083333}$ (2)

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

 > $A≔\mathrm{Sample}\left('\mathrm{Β}'\left(3,5\right),{10}^{5}\right):$
 > $\mathrm{CentralMoment}\left(A,3\right)$
 ${0.00134022023133361}$ (3)

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

 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5,2\right)\right):$
 > $B≔\mathrm{Sample}\left(X,{10}^{6}\right):$
 > $\left[\mathrm{CentralMoment}\left(B,4\right),\mathrm{StandardError}\left(\mathrm{CentralMoment},B,4\right)\right]$
 $\left[{48.0775651884354}{,}{0.158051691104348302}\right]$ (4)

Create a beta-distributed random variable $Y$ and compute the third central moment of $\frac{1}{Y+2}$.

 > $Y≔\mathrm{RandomVariable}\left('\mathrm{Β}'\left(5,2\right)\right):$
 > $\mathrm{CentralMoment}\left(\frac{1}{Y+2},3,\mathrm{numeric}\right)$
 ${0.00001053304160}$ (5)

Verify this using simulation.

 > $C≔\mathrm{Sample}\left(\frac{1}{Y+2},{10}^{5}\right):$
 > $\mathrm{CentralMoment}\left(C,3\right)$
 ${0.0000106403819721152}$ (6)

Compute the average central moment of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{CentralMoment}\left(V,4,\mathrm{weights}=W\right)$
 ${Float}{}\left({\mathrm{undefined}}\right)$ (7)
 > $\mathrm{CentralMoment}\left(V,4,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)$
 ${137.689183427122}$ (8)

Consider the following Matrix data set.

 > $M≔\mathrm{Matrix}\left(\left[\left[3,1130,114694\right],\left[4,1527,127368\right],\left[3,907,88464\right],\left[2,878,96484\right],\left[4,995,128007\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{3}& {1130}& {114694}\\ {4}& {1527}& {127368}\\ {3}& {907}& {88464}\\ {2}& {878}& {96484}\\ {4}& {995}& {128007}\end{array}\right]$ (9)

We compute the third central moment of each column.

 > $\mathrm{CentralMoment}\left(M,3\right)$
 $\left[\begin{array}{ccc}{-0.144000000000000}& {1.38374995680000}{}{{10}}^{{7}}& {-1.03252293646843}{}{{10}}^{{12}}\end{array}\right]$ (10)

References

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

Compatibility

 • The M parameter was introduced in Maple 16.