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Student[Statistics]

  

Kurtosis

  

compute the coefficient of kurtosis

 

Calling Sequence

Parameters

Description

Computation

Examples

References

Compatibility

Calling Sequence

Kurtosis(A, numeric_option)

Kurtosis(M, numeric_option)

Kurtosis(X, numeric_option, inert_option)

Parameters

A

-

data sample

M

-

Matrix data sample

X

-

algebraic; random variable

numeric_option

-

(optional) equation of the form numeric=value where value is true or false

inert_option

-

(optional) equation of the form inert=value where value is true or false

Description

• 

The Kurtosis function computes the coefficient of kurtosis of the specified random variable or data sample. In the data sample case, the following formula for the kurtosis is used:

KurtosisA=NMomentA,4,origin=MeanAN1VarianceA2,

  

where N is the number of elements in A. In the random variable case, Maple uses the limit of that formula for expr, that is,

  

KurtosisX=MomentX,4,origin=MeanXVarianceX2.

• 

There is a different quantity that some authors call kurtosis. This quantity is called excess kurtosis here. The excess kurtosis is not predefined in Maple, but it can be easily obtained by subtracting 3 from the kurtosis: ExcessKurtosisKurtosis3.

• 

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

• 

If the option inert is not included or is specified to be inert=false, then the function will return the actual value of the result. If inert or inert=true is specified, then the function will return the formula of evaluating the actual value.

Computation

• 

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

• 

If there are floating point values or the option numeric is included, then the computation is done in floating point. Otherwise the computation is exact.

• 

By default, the kurtosis is computed according to the rules mentioned above. To always compute the kurtosis numerically, specify the numeric or numeric = true option.

Examples

withStudent[Statistics]:

Compute the coefficient of kurtosis of the log normal distribution with parameters μ and σ.

KurtosisLogNormalRandomVariableμ,σ

ⅇ8σ2+4μ4ⅇ5σ2+4μ+6ⅇ3σ2+4μ3ⅇ2σ2+4μⅇσ2+2μ2ⅇσ212

(1)

Use numeric parameters for the beta distribution.

KurtosisBetaRandomVariable3,5

711275

(2)

KurtosisBetaRandomVariable3,5,numeric

2.585454546

(3)

Use the inert option.

KurtosisBetaRandomVariable3,5,inert

∫01105_t2+∫01105_t131+_t14ⅆ_t14_t221+_t24ⅆ_t2∫01105_t0+∫01105_t31+_t4ⅆ_t2_t021+_t04ⅆ_t02

(4)

evalfKurtosisBetaRandomVariable3,5,inert

2.585454545

(5)

Consider the following list of data.

A1,2,π,ⅇ1.5,3

A1,2,π,4.481689070,3

(6)

KurtosisA

1.92292561031128

(7)

Consider the following Matrix data set.

MMatrix3,1,11,4,1.5,28,3,ln3,31,2,0,4,4,9.2,7

M311141.5283ln33120449.27

(8)

We compute the kurtosis of each of the columns.

KurtosisM

3622452.519272430263041217684211966045

(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 Student[Statistics][Kurtosis] command was introduced in Maple 18.

• 

For more information on Maple 18 changes, see Updates in Maple 18.

See Also

kurtosis

Statistics[Kurtosis]

Student

Student[Statistics]

Student[Statistics][RandomVariable]