Kurtosis - Maple Help

Student[Statistics]

 Kurtosis
 compute the coefficient of kurtosis

 Calling Sequence Kurtosis(A, numeric_option) Kurtosis(M, numeric_option) Kurtosis(X, numeric_option, inert_option)

Parameters

 A - M - 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:

$\mathrm{Kurtosis}\left(A\right)=\frac{N\mathrm{Moment}\left(A,4,\mathrm{origin}=\mathrm{Mean}\left(A\right)\right)}{\left(N-1\right){\mathrm{Variance}\left(A\right)}^{2}},$

 where N is the number of elements in A. In the random variable case, Maple uses the limit of that formula for $N↦\mathrm{\infty }$, that is,
 $\mathrm{Kurtosis}\left(X\right)=\frac{\mathrm{Moment}\left(X,4,\mathrm{origin}=\mathrm{Mean}\left(X\right)\right)}{{\mathrm{Variance}\left(X\right)}^{2}}$.
 • 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: $\mathrm{ExcessKurtosis}≔\mathrm{Kurtosis}-3$.
 • 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

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

Compute the coefficient of kurtosis of the log normal distribution with parameters $\mathrm{\mu }$ and $\mathrm{\sigma }$.

 > $\mathrm{Kurtosis}\left(\mathrm{LogNormalRandomVariable}\left(\mathrm{μ},\mathrm{σ}\right)\right)$
 $\frac{{6}{}{{ⅇ}}^{{3}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{\mathrm{\mu }}}{+}{{ⅇ}}^{{8}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{\mathrm{\mu }}}{-}{3}{}{{ⅇ}}^{{2}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{\mathrm{\mu }}}{-}{4}{}{{ⅇ}}^{{5}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{\mathrm{\mu }}}}{{\left({{ⅇ}}^{{{\mathrm{\sigma }}}^{{2}}{+}{2}{}{\mathrm{\mu }}}\right)}^{{2}}{}{\left({{ⅇ}}^{{{\mathrm{\sigma }}}^{{2}}}{-}{1}\right)}^{{2}}}$ (1)

Use numeric parameters for the beta distribution.

 > $\mathrm{Kurtosis}\left(\mathrm{BetaRandomVariable}\left(3,5\right)\right)$
 $\frac{{711}}{{275}}$ (2)
 > $\mathrm{Kurtosis}\left(\mathrm{BetaRandomVariable}\left(3,5\right),\mathrm{numeric}\right)$
 ${2.585454546}$ (3)

Use the inert option.

 > $\mathrm{Kurtosis}\left(\mathrm{BetaRandomVariable}\left(3,5\right),\mathrm{inert}\right)$
 $\frac{{{\int }}_{{0}}^{{1}}{105}{}{\left({-}{\mathrm{_t2}}{+}{{\int }}_{{0}}^{{1}}{105}{}{{\mathrm{_t1}}}^{{3}}{}{\left({-}{1}{+}{\mathrm{_t1}}\right)}^{{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t1}}\right)}^{{4}}{}{{\mathrm{_t2}}}^{{2}}{}{\left({-}{1}{+}{\mathrm{_t2}}\right)}^{{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t2}}}{{\left({{\int }}_{{0}}^{{1}}{105}{}{\left({-}{\mathrm{_t0}}{+}{{\int }}_{{0}}^{{1}}{105}{}{{\mathrm{_t}}}^{{3}}{}{\left({-}{1}{+}{\mathrm{_t}}\right)}^{{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t}}\right)}^{{2}}{}{{\mathrm{_t0}}}^{{2}}{}{\left({-}{1}{+}{\mathrm{_t0}}\right)}^{{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t0}}\right)}^{{2}}}$ (4)
 > $\mathrm{evalf}\left(\mathrm{Kurtosis}\left(\mathrm{BetaRandomVariable}\left(3,5\right),\mathrm{inert}\right)\right)$
 ${2.585454545}$ (5)

Consider the following list of data.

 > $A≔\left[1,2,\mathrm{Pi},{ⅇ}^{1.5},-3\right]$
 ${A}{≔}\left[{1}{,}{2}{,}{\mathrm{\pi }}{,}{4.481689070}{,}{-3}\right]$ (6)
 > $\mathrm{Kurtosis}\left(A\right)$
 ${1.92292561031128}$ (7)

Consider the following Matrix data set.

 > $M≔\mathrm{Matrix}\left(\left[\left[3,1,11\right],\left[4,1.5,28\right],\left[3,\mathrm{ln}\left(3\right),31\right],\left[2,0,4\right],\left[4,9.2,7\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{3}& {1}& {11}\\ {4}& {1.5}& {28}\\ {3}& {\mathrm{ln}}{}\left({3}\right)& {31}\\ {2}& {0}& {4}\\ {4}& {9.2}& {7}\end{array}\right]$ (8)

We compute the kurtosis of each of the columns.

 > $\mathrm{Kurtosis}\left(M\right)$
 $\left[\begin{array}{ccc}\frac{{362}}{{245}}& {2.51927243026304}& \frac{{12176842}}{{11966045}}\end{array}\right]$ (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.