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Distributed—Wolfram Documentation

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Distributed[x,dist]

or xdist asserts that the random variable x is distributed according to the probability distribution dist.

Distributed[{x1,x2,…},dist]

or {x1,x2,…}dist asserts that the random vector {x1,x2,…} is distributed according to the multivariate probability distribution dist.

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Distributed

Distributed[x,dist]

or xdist asserts that the random variable x is distributed according to the probability distribution dist.

Distributed[{x1,x2,…},dist]

or {x1,x2,…}dist asserts that the random vector {x1,x2,…} is distributed according to the multivariate probability distribution dist.

Details

  • xdist can be entered as x dist dist x \[Distributed] dist .
  • xdist is a symbolic object used in functions such as Probability and Expectation.
  • xproc[t] asserts that the random variable x is distributed according to the distribution SliceDistribution[proc,t] for the random process proc.
  • The probability distribution dist can be any symbolic probability distribution specification.

Examples

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Basic Examples  (3)

Compute the probability of an event in a symbolic probability distribution:

Wolfram Language code: Probability[x ^ 2 < 5, xNormalDistribution[]]
Wolfram Language code: N[%]

Obtain the numerical value directly using NProbability:

Wolfram Language code: NProbability[x ^ 2 < 5, xNormalDistribution[]]

Compute the expectation of a function in a multivariate probability distribution:

Wolfram Language code: Expectation[x ^ 2 + 3E ^ (y), {x, y}BinormalDistribution[1 / 3]]
Wolfram Language code: N[%]
Wolfram Language code: NExpectation[x ^ 2 + 3E ^ (y), {x, y}BinormalDistribution[1 / 3]]

Mean and variance for the distribution obtained by transformation of a random variable:

Wolfram Language code: dist = TransformedDistribution[3x + 5, xPoissonDistribution[2]];
Wolfram Language code: Mean[dist]
Wolfram Language code: Variance[dist]

Scope  (5)

Compute an expectation for a parametric distribution:

Wolfram Language code: Expectation[x ^ 2 + 3x, xExponentialDistribution[2]]

Derived distribution:

Wolfram Language code: Expectation[x ^ 2, xTransformedDistribution[x ^ 3, xExponentialDistribution[2]]]

Formula distribution:

Wolfram Language code: Expectation[x ^ 2, xProbabilityDistribution[Piecewise[{{x ^ 2 / 9, 0 < x ≤ 3}}], {x, -∞, ∞}]]

Data distribution:

Wolfram Language code: data = RandomVariate[NormalDistribution[], 10^4];
Wolfram Language code: 𝒟 = HistogramDistribution[data];
Wolfram Language code: Expectation[x ^ 2, x𝒟]

Use implicit time to compute a probabiity for a random process:

Wolfram Language code: NProbability[x[1] > 7, xPoissonProcess[4]]

Obtain the same result using the corresponding slice distribution:

Wolfram Language code: NProbability[x > 7, xPoissonProcess[4][1]]

Compute an expectation for a multivariate slice of a random process:

Wolfram Language code: Expectation[E ^ (-x[4] + x[5]), xWienerProcess[]]

Define a TransformedProcess:

Wolfram Language code: 𝒫 = TransformedProcess[b[2t + 1], bWienerProcess[], t];

Compute an expectation for a time slice of the process:

Wolfram Language code: Expectation[x[3], x𝒫]

Simulate a TransformedProcess:

Wolfram Language code: 𝒫 = TransformedProcess[b[2t + 1], bWienerProcess[], t];
Wolfram Language code: ListLinePlot[RandomFunction[𝒫, {0, 10, 0.1}, 3]]

Generalizations & Extensions  (1)

Compute a probability for a distribution specified as a list:

Wolfram Language code: r = RandomVariate[NormalDistribution[], 10 ^ 5];
Wolfram Language code: Probability[x ^ 2 < 1, xr]//N

Compare with the probability using the distribution itself:

Wolfram Language code: NProbability[x ^ 2 < 1, xNormalDistribution[]]
Wolfram Research (2010), Distributed, Wolfram Language function, https://reference.wolfram.com/language/ref/Distributed.html.

Text

Wolfram Research (2010), Distributed, Wolfram Language function, https://reference.wolfram.com/language/ref/Distributed.html.

CMS

Wolfram Language. 2010. "Distributed." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Distributed.html.

APA

Wolfram Language. (2010). Distributed. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Distributed.html

BibTeX

@misc{reference.wolfram_2026_distributed, author="Wolfram Research", title="{Distributed}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/Distributed.html}", note=[Accessed: 14-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_distributed, organization={Wolfram Research}, title={Distributed}, year={2010}, url={https://reference.wolfram.com/language/ref/Distributed.html}, note=[Accessed: 14-November-2026]}