({{, (RefLink[Distributed, paclet:ref/Distributed])[x,dist] 
or x\[Distributed]dist asserts that the random variable x is distributed according to the probability distribution dist.}, {, (RefLink[Distributed, paclet:ref/Distributed])[{(Subscript[x, 1]),(Subscript[x, 2]),…},dist]
or {(Subscript[x, 1]),(Subscript[x, 2]),…}\[Distributed]dist asserts that the random vector {(Subscript[x, 1]),(Subscript[x, 2]),…} is distributed according to the multivariate probability distribution dist.}})

OpenersNotesSection(Spacer1)Details

x\[Distributed]dist can be entered as x (Key0[esc]) dist (Key0[esc]) dist or x∖[Distributed]dist.

x\[Distributed]dist is a symbolic object used in functions such as (RefLink[Probability, paclet:ref/Probability]) and (RefLink[Expectation, paclet:ref/Expectation]).

x\[Distributed]proc[t] asserts that the random variable x is distributed according to the distribution (RefLink[SliceDistribution, paclet:ref/SliceDistribution])[proc,t] for the random process proc.

The probability distribution dist can be any symbolic probability distribution specification.

OpenersPrimaryExamplesSection(Spacer1)Examples  (9)

OpenersExampleSection0(Spacer1)Basic Examples  (3)

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

Probability[x^2<5,x\[Distributed]NormalDistribution[]]

N[%]

Obtain the numerical value directly using (RefLink[NProbability, paclet:ref/NProbability]):

NProbability[x^2<5,x\[Distributed]NormalDistribution[]]

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

Expectation[x^2+3E^(y),{x,y}\[Distributed]BinormalDistribution[1/3]]

N[%]

NExpectation[x^2+3E^(y),{x,y}\[Distributed]BinormalDistribution[1/3]]

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

dist=TransformedDistribution[3x+5,x\[Distributed]PoissonDistribution[2]];

Mean[dist]

Variance[dist]

OpenersExampleSection1(Spacer1)Scope  (5)

Compute an expectation for a parametric distribution:

Expectation[x^2+3x,x\[Distributed]ExponentialDistribution[2]]

Derived distribution:

Expectation[x^2,x\[Distributed]TransformedDistribution[x^3,x\[Distributed]ExponentialDistribution[2]]]

Formula distribution:

Expectation[x^2,x\[Distributed]ProbabilityDistribution[Piecewise[{{x^2/9,0<x≤3}}],{x,-∞,∞}]]

Data distribution:

data=RandomVariate[NormalDistribution[],(10^4)];

𝒟=HistogramDistribution[data];

Expectation[x^2,x\[Distributed]𝒟]

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

NProbability[x[1]>7,x\[Distributed]PoissonProcess[4]]

Obtain the same result using the corresponding slice distribution:

NProbability[x>7,x\[Distributed]PoissonProcess[4][1]]

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

Expectation[E^(-x[4]+x[5]),x\[Distributed]WienerProcess[]]

Define a (RefLink[TransformedProcess, paclet:ref/TransformedProcess]):

𝒫=TransformedProcess[b[2t+1],b\[Distributed]WienerProcess[],t];

Compute an expectation for a time slice of the process:

Expectation[x[3],x\[Distributed]𝒫]

Simulate a (RefLink[TransformedProcess, paclet:ref/TransformedProcess]):

𝒫=TransformedProcess[b[2t+1],b\[Distributed]WienerProcess[],t];

ListLinePlot[RandomFunction[𝒫,{0,10,0.1},3]]

OpenersExampleSection2(Spacer1)Generalizations & Extensions  (1)

Compute a probability for a distribution specified as a list:

r=RandomVariate[NormalDistribution[],10^5];

Probability[x^2<1,x\[Distributed]r]//N

Compare with the probability using the distribution itself:

NProbability[x^2<1,x\[Distributed]NormalDistribution[]]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Probability, paclet:ref/Probability])  ▪  (RefLink[NProbability, paclet:ref/NProbability])  ▪  (RefLink[Expectation, paclet:ref/Expectation])  ▪  (RefLink[NExpectation, paclet:ref/NExpectation])  ▪  (RefLink[TransformedDistribution, paclet:ref/TransformedDistribution])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Probability & Statistics, paclet:guide/ProbabilityAndStatistics])}, {▪(RefLinkPlain[Random Variables, paclet:guide/RandomVariables])}, {▪(RefLinkPlain[Wolfram Language Syntax, paclet:guide/Syntax])}, {▪(RefLinkPlain[Random Graphs, paclet:guide/RandomGraphs])}})}})