For a data set A (given as e.g. a Vector) or a Matrix data set M, the ExpectedValue function computes the expected value of f with respect to the sample distribution of A or of the columns of M, respectively.

For a random variable X the ExpectedValue command computes the expected value of f(X). If X is an expression involving random variables, then the expected value of X is computed.

The first parameter X is a random variable or an algebraic expression involving random variables.

The second parameter is a function.

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

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.

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

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 ExpectedValue command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the ExpectedValue 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 $1$.

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 expected value is computed using exact arithmetic. To compute the expected value numerically, specify the numeric or numeric = true option.

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

$X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(a\,b\right)\right)\:$

$Y≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(c\,d\right)\right)\:$

$\mathrm{ExpectedValue}\left(X^2\right)$

$a^2+b^2$

$\mathrm{ExpectedValue}\left(X+Y\right)$

$a+c$

$\mathrm{ExpectedValue}\left({\left(X-a\right)}^2\right)$

$b^2$

$Z≔\mathrm{RandomVariable}\left(\mathrm{Exponential}\left(2\right)\right)\:$

$A≔\mathrm{Sample}\left(Z\,{10}^5\right)\:$

$\mathrm{ExpectedValue}\left(Z^2\right)$

$8$

$\mathrm{ExpectedValue}\left(A\,t\mapsto t^2\right)$

$8.00071728582744$

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

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

$\mathrm{ExpectedValue}\left(M\,\ln \right)$

$\left[\begin{array}{ccc}1.13259209602719& 6.97031299956677& 11.6064364945012\end{array}\right]$

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

The M parameter was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.