Statistics - Maple Help

Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Quantities : Statistics/ExpectedValue

Statistics

 ExpectedValue
 compute expected values

 Calling Sequence ExpectedValue(A, f, ds_options) ExpectedValue(M, f, ds_options) ExpectedValue(X, f, rv_options) ExpectedValue(X, rv_options)

Parameters

 A - M - X - algebraic; distribution, random variable f - operator; any function ds_options - (optional) equation(s) of the form option=value where option is one of ignore, or weights; specify options for computing the expected value of a data set rv_options - (optional) equation of the form numeric=value; specifies options for computing the expected value of a random variable

Description

 • 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.

Computation

 • 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.

Data Set Options

 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 $1$.

Random Variable Options

 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.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(a,b\right)\right):$
 > $Y≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(c,d\right)\right):$
 > $\mathrm{ExpectedValue}\left({X}^{2}\right)$
 ${{a}}^{{2}}{+}{{b}}^{{2}}$ (1)
 > $\mathrm{ExpectedValue}\left(X+Y\right)$
 ${a}{+}{c}$ (2)
 > $\mathrm{ExpectedValue}\left({\left(X-a\right)}^{2}\right)$
 ${{b}}^{{2}}$ (3)
 > $Z≔\mathrm{RandomVariable}\left(\mathrm{Exponential}\left(2\right)\right):$
 > $A≔\mathrm{Sample}\left(Z,{10}^{5}\right):$
 > $\mathrm{ExpectedValue}\left({Z}^{2}\right)$
 ${8}$ (4)
 > $\mathrm{ExpectedValue}\left(A,t→{t}^{2}\right)$
 ${8.00071728582744}$ (5)

Consider the following Matrix data set.

 > $M≔\mathrm{Matrix}\left(\left[\left[3,1130,114694\right],\left[4,1527,127368\right],\left[3,907,88464\right],\left[2,878,96484\right],\left[4,995,128007\right]\right]\right)$
 ${M}{:=}\left[\begin{array}{rrr}{3}& {1130}& {114694}\\ {4}& {1527}& {127368}\\ {3}& {907}& {88464}\\ {2}& {878}& {96484}\\ {4}& {995}& {128007}\end{array}\right]$ (6)

We compute the expected value of the natural logarithm of each of the column data sets.

 > $\mathrm{ExpectedValue}\left(M,\mathrm{ln}\right)$
 $\left[\begin{array}{ccc}{1.13259209602719}& {6.97031299956677}& {11.6064364945012}\end{array}\right]$ (7)

References

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

Compatibility

 • The M parameter was introduced in Maple 16.