BUILT-IN MATHEMATICA 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:

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Compute the expectation of an arbitrary expression:

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Compute a conditional expectation:

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### Properties & Relations (9)Properties & Relations (9)

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