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ExpectedValue[f,list]

gives the expected value of the pure function f with respect to the values in list.

ExpectedValue[f,list,x]

gives the expected value of the function f of x with respect to the values of list.

ExpectedValue[f,dist]

gives the expected value of the pure function f with respect to the symbolic distribution dist.

ExpectedValue[f,dist,x]

gives the expected value of the function f of x with respect to the symbolic distribution dist.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Assumptions  
Applications  
Properties & Relations  
See Also
Tech Notes
History
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ExpectedValue

As of Version 8.0, ExpectedValue has been superseded by Expectation and NExpectation.

ExpectedValue[f,list]

gives the expected value of the pure function f with respect to the values in list.

ExpectedValue[f,list,x]

gives the expected value of the function f of x with respect to the values of list.

ExpectedValue[f,dist]

gives the expected value of the pure function f with respect to the symbolic distribution dist.

ExpectedValue[f,dist,x]

gives the expected value of the function f of x with respect to the symbolic distribution dist.

Details and Options

  • For the list , the expected value of f is given by .
  • For a continuous distribution dist, the expected value of f 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 expected value of f is given by where is the probability mass function of dist and summation is over the domain of dist.
  • The following option can be given:
  • Assumptions $Assumptionsassumptions to make about parameters

Examples

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Basic Examples  (3)

Find the expected value of in a Poisson distribution:

Wolfram Language code: ExpectedValue[x ^ 4, PoissonDistribution[μ], x]

Use a pure function:

Wolfram Language code: ExpectedValue[# ^ 4&, PoissonDistribution[μ]]

Expected value for a list:

Wolfram Language code: ExpectedValue[Sin, {1, 4, 2, 5, 3, 7, 2}]

Scope  (3)

Compute the expected value of any function:

Wolfram Language code: ExpectedValue[x ^ 4 Exp[-Abs[x]], NormalDistribution[0, 1], x]

Do the computation numerically:

Wolfram Language code: ExpectedValue[x ^ 4 Exp[-Abs[x]], NormalDistribution[0, 1.], x]

Obtain expectations with conditions:

Wolfram Language code: ExpectedValue[x ^ n, CauchyDistribution[0, 1], x]
Wolfram Language code: ExpectedValue[x ^ n, CauchyDistribution[0, 1], x, Assumptions -> -1 < n < 1]

Options  (1)

Assumptions  (1)

Obtain results correct for given assumptions on symbols:

Wolfram Language code: ExpectedValue[x ^ n, ParetoDistribution[k, α], x]
Wolfram Language code: ExpectedValue[x ^ n, ParetoDistribution[k, α], x, Assumptions -> α > n]

Applications  (2)

Obtain the raw moments of a distribution:

Wolfram Language code: ExpectedValue[x ^ n, ChiSquareDistribution[ν], x, Assumptions -> n > 0]

Construct a mixture density, here a Poisson‐inverse Gaussian mixture:

Wolfram Language code: ExpectedValue[PDF[PoissonDistribution[μ], k], InverseGaussianDistribution[m, λ], μ, Assumptions -> k > 0]

Properties & Relations  (7)

ExpectedValue of a function is the integral or sum of that function times the PDF:

Wolfram Language code: ExpectedValue[x ^ 3, ExponentialDistribution[λ], x]
Wolfram Language code: Integrate[x ^ 3 PDF[ExponentialDistribution[λ], x], {x, 0, Infinity}, Assumptions -> λ > 0]
Wolfram Language code: ExpectedValue[Exp[k], BinomialDistribution[n, p], k]
Wolfram Language code: Sum[Exp[k] PDF[BinomialDistribution[n, p], k], {k, 0, n}]

ExpectedValue of for real t is the CharacteristicFunction:

Wolfram Language code: ExpectedValue[Exp[I t x], GumbelDistribution[α, β], x, Assumptions -> Element[t, Reals]]
Wolfram Language code: CharacteristicFunction[GumbelDistribution[α, β], t]

ExpectedValue of a constant is the constant:

Wolfram Language code: ExpectedValue[5, BetaDistribution[α, β], x]

ExpectedValue of a random variable is the Mean:

Wolfram Language code: ExpectedValue[x, ChiSquareDistribution[ν], x]
Wolfram Language code: Mean[ChiSquareDistribution[ν]]

ExpectedValue of the squared difference from the Mean is the Variance:

Wolfram Language code: ExpectedValue[(x - ν) ^ 2, ChiSquareDistribution[ν], x]
Wolfram Language code: Variance[ChiSquareDistribution[ν]]

ExpectedValue for a list is a Mean:

Wolfram Language code: ExpectedValue[f, Range[5]]
Wolfram Language code: Mean[Map[f, Range[5]]]

CentralMoment is equivalent to an expected value:

Wolfram Language code: list = Range[100];
Wolfram Language code: ExpectedValue[(x - Mean[list]) ^ 5, list, x] == CentralMoment[list, 5]
Wolfram Research (2007), ExpectedValue, Wolfram Language function, https://reference.wolfram.com/language/ref/ExpectedValue.html (updated 2008).

Text

Wolfram Research (2007), ExpectedValue, Wolfram Language function, https://reference.wolfram.com/language/ref/ExpectedValue.html (updated 2008).

CMS

Wolfram Language. 2007. "ExpectedValue." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/ExpectedValue.html.

APA

Wolfram Language. (2007). ExpectedValue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ExpectedValue.html

BibTeX

@misc{reference.wolfram_2026_expectedvalue, author="Wolfram Research", title="{ExpectedValue}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/ExpectedValue.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_expectedvalue, organization={Wolfram Research}, title={ExpectedValue}, year={2008}, url={https://reference.wolfram.com/language/ref/ExpectedValue.html}, note=[Accessed: 13-June-2026]}