The Moment function computes the moment of order n of the specified random variable or data sample.

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]).

The second parameter can be any algebraic expression.

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 data sample, 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.

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).

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

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

$\mathrm{Moment}\left(\mathrm{BetaRandomVariable}\left(4\,7\right)\,4\right)$

$\frac{5}{143}$

$\mathrm{Moment}\left(\mathrm{BetaRandomVariable}\left(4\,7\right)\,4\,\mathrm{numeric}\right)$

$0.03496503497$

$\mathrm{Moment}\left(\mathrm{BetaRandomVariable}\left(4\,7\right)\,2\,\mathrm{origin}=\frac{1}{2}\right)$

$\frac{5}{132}$

$\mathrm{Moment}\left(\mathrm{BetaRandomVariable}\left(4\,7\right)\,2\,\mathrm{origin}=\frac{1}{2}\,\mathrm{numeric}\right)$

$0.03787878788$

$\mathrm{Moment}\left(\mathrm{BetaRandomVariable}\left(4\,7\right)\,a\,\mathrm{origin}=1\right)$

$\frac{5040{\left(\mathrm{-1}\right)}^a}{\left(a+7\right)\left(8+a\right)\left(9+a\right)\left(a+10\right)}$

$\mathrm{Moment}\left(\mathrm{BetaRandomVariable}\left(4\,7\right)\,2\,\mathrm{origin}=b\right)$

$\frac{5}{33}-\frac{8}{11}b+b^2$

$\mathrm{Moment}\left(\mathrm{BetaRandomVariable}\left(4\,7\right)\,2\,\mathrm{origin}=\frac{1}{2}\,\mathrm{inert}\right)$

${\int }_0^{1}840{\left(\mathrm{\_t1}-\frac{1}{2}\right)}^2{\mathrm{\_t1}}^3{\left(1-\mathrm{\_t1}\right)}^6\ⅆ\mathrm{\_t1}$

$\mathrm{evalf}\left(\mathrm{Moment}\left(\mathrm{BetaRandomVariable}\left(4\,7\right)\,2\,\mathrm{origin}=\frac{1}{2}\,\mathrm{inert}\right)\right)$

$0.03787878788$

$X≔\mathrm{ExponentialRandomVariable}\left(2\right)\:$

$\mathrm{Moment}\left(\frac{1}{X+2}\,3\,\mathrm{numeric}\right)$

$0.03727171015$

$A≔⟨0\,2\,3\,5\,2\,\mathrm{\pi }\,2\,-4\,\ln \left(2\right)⟩$

$A≔\left[\begin{array}{c}0\\ 2\\ 3\\ 5\\ 2\\ \mathrm{\pi }\\ 2\\ \mathrm{-4}\\ \ln \left(2\right)\end{array}\right]$

$\mathrm{Moment}\left(A\,2\,\mathrm{origin}=3\right)$

$\frac{65}{9}+\frac{{\left(\mathrm{\pi }-3\right)}^2}{9}+\frac{{\left(\ln \left(2\right)-3\right)}^2}{9}$

$M≔\mathrm{Matrix}\left(\left[\left[3\,\mathrm{\pi }\,114\right]\,\left[5.0\,3\,-12\right]\,\left[\ln \left(4\right)\,\ln \left(5\right)\,88\right]\,\left[2\,5\,54\right]\,\left[4.2\,23\,17\right]\right]\right)$

$M≔\left[\begin{array}{ccc}3& \mathrm{\pi }& 114\\ 5.0& 3& \mathrm{-12}\\ 2\ln \left(2\right)& \ln \left(5\right)& 88\\ 2& 5& 54\\ 4.2& 23& 17\end{array}\right]$

$\mathrm{Moment}\left(M\,2\right)$

$\left[\begin{array}{ccc}11.51236241& \frac{563}{5}+\frac{{\mathrm{\pi }}^2}{5}+\frac{{\ln \left(5\right)}^2}{5}& \frac{24089}{5}\end{array}\right]$

$\mathrm{Moment}\left(M\,2\,'\mathrm{origin}=3'\right)$

$\left[\begin{array}{ccc}1.808809178& \frac{404}{5}+\frac{{\left(\mathrm{\pi }-3\right)}^2}{5}+\frac{{\left(\ln \left(5\right)-3\right)}^2}{5}& \frac{22568}{5}\end{array}\right]$

$\mathrm{Moment}\left(M\,\left[2\,3\,1\right]\,'\mathrm{origin}=\left[3\,100\,85\right]'\right)$

$\left[\begin{array}{ccc}1.808809178& -\frac{2226581}{5}+\frac{{\left(\mathrm{\pi }-100\right)}^3}{5}+\frac{{\left(\ln \left(5\right)-100\right)}^3}{5}& -\frac{164}{5}\end{array}\right]$

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

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

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