# Wolfram Language & System 10.4 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# Expectation

Expectation[expr,xdist]
gives the expectation of expr under the assumption that x follows the probability distribution dist.

Expectation[expr,xdata]
gives the expectation of expr under the assumption that x follows the probability distribution given by data.

Expectation[expr,{x1,x2,}dist]
gives the expectation of expr under the assumption that follows the multivariate distribution dist.

Expectation[expr,{x1dist1,x2dist2,}]
gives the expectation of expr under the assumption that , , are independent and follow the distributions , , .

Expectation[exprpred,]
gives the conditional expectation of expr given pred.

## Details and OptionsDetails and Options

• Expectation is also known as expected value.
• can be entered as x EscdistEsc dist or .
• can be entered as expr EsccondEsc pred or .
• For a continuous distribution dist, the expectation of expr is given by where is the probability density function of dist and the integral is taken over the domain of dist.
• For a discrete distribution dist, the probability of expr is given by where is the probability density function of dist and the summation is taken over the domain of dist.
• For a dataset data, the expectation of expr is given by Sum[expr,{x,data}]/Length[data].
• Univariate data is given as a list of values and multivariate data is given as a list of vectors .
• Expectation[expr,{x1dist1,x2dist2}] corresponds to Expectation[Expectation[expr,x2dist2],x1dist1] so that the last variable is summed or integrated first.
• N[Expectation[]] calls NExpectation for expectations that cannot be done symbolically.
• The following options can be given:
•  Assumptions \$Assumptions assumptions to make about parameters GenerateConditions False whether to generate conditions on parameters Method Automatic what method to use

## ExamplesExamplesopen allclose all

### Basic Examples  (3)Basic Examples  (3)

Compute the expectation of a polynomial expression:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=
 In[3]:=
 Out[3]=
 In[4]:=
 Out[4]=

Compute the expectation of an arbitrary expression:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=
 In[3]:=
 Out[3]=

Compute a conditional expectation:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=