Expectation

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

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

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

Expectation[expr, {x1Distributeddist1, x2Distributeddist2, ...}]
gives the expectation of expr under the assumption that , , ... are independent and follow the distributions , , ....

Expectation[exprConditionedpred, ...]
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, {x1Distributeddist1, x2Distributeddist2}] corresponds to Expectation[Expectation[expr, x2Distributeddist2], x1Distributeddist1] 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$Assumptionsassumptions to make about parameters
    GenerateConditionsFalsewhether to generate conditions on parameters
    MethodAutomaticwhat 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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