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:

where N is the number of elements in A. In the random variable case, Maple uses the limit of that formula for $N\mapsto \mathrm{\infty }$, 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.

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.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Statistics}\right]\right)\:$

$\mathrm{Kurtosis}\left(\mathrm{LogNormalRandomVariable}\left(\mathrm{\mu }\,\mathrm{\sigma }\right)\right)$

$\frac{{\ⅇ}^{8{\mathrm{\sigma }}^2+4\mathrm{\mu }}-3{\ⅇ}^{2{\mathrm{\sigma }}^2+4\mathrm{\mu }}-4{\ⅇ}^{5{\mathrm{\sigma }}^2+4\mathrm{\mu }}+6{\ⅇ}^{3{\mathrm{\sigma }}^2+4\mathrm{\mu }}}{{\left({\ⅇ}^{{\mathrm{\sigma }}^2+2\mathrm{\mu }}\right)}^2{\left({\ⅇ}^{{\mathrm{\sigma }}^2}-1\right)}^2}$

$\mathrm{Kurtosis}\left(\mathrm{BetaRandomVariable}\left(3\,5\right)\right)$

$\frac{711}{275}$

$\mathrm{Kurtosis}\left(\mathrm{BetaRandomVariable}\left(3\,5\right)\,\mathrm{numeric}\right)$

$2.585454546$

$\mathrm{Kurtosis}\left(\mathrm{BetaRandomVariable}\left(3\,5\right)\,\mathrm{inert}\right)$

$\frac{{\int }_0^{1}105{\left(-\mathrm{\_t2}+{\int }_0^{1}105{\mathrm{\_t1}}^3{\left(-1+\mathrm{\_t1}\right)}^4\ⅆ\mathrm{\_t1}\right)}^4{\mathrm{\_t2}}^2{\left(-1+\mathrm{\_t2}\right)}^4\ⅆ\mathrm{\_t2}}{{\left({\int }_0^{1}105{\left(-\mathrm{\_t0}+{\int }_0^{1}105{\mathrm{\_t}}^3{\left(-1+\mathrm{\_t}\right)}^4\ⅆ\mathrm{\_t}\right)}^2{\mathrm{\_t0}}^2{\left(-1+\mathrm{\_t0}\right)}^4\ⅆ\mathrm{\_t0}\right)}^2}$

$\mathrm{evalf}\left(\mathrm{Kurtosis}\left(\mathrm{BetaRandomVariable}\left(3\,5\right)\,\mathrm{inert}\right)\right)$

$2.585454545$

$A≔\left[1\,2\,\mathrm{\pi }\,\exp \left(1.5\right)\,-3\right]$

$A≔\left[1\,2\,\mathrm{\pi }\,4.481689070\,\mathrm{-3}\right]$

$\mathrm{Kurtosis}\left(A\right)$

$1.92292561031128$

$M≔\mathrm{Matrix}\left(\left[\left[3\,1\,11\right]\,\left[4\,1.5\,28\right]\,\left[3\,\ln \left(3\right)\,31\right]\,\left[2\,0\,4\right]\,\left[4\,9.2\,7\right]\right]\right)$

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

$\mathrm{Kurtosis}\left(M\right)$

$\left[\begin{array}{ccc}\frac{362}{245}& 2.51927243026304& \frac{12176842}{11966045}\end{array}\right]$

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

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

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