({{, (RefLink[SliceDistribution, paclet:ref/SliceDistribution])[proc,t] 
represents the distribution of the process state at time t.}, {, (RefLink[SliceDistribution, paclet:ref/SliceDistribution])[proc,{(Subscript[t, 1]),…,(Subscript[t, k])}]
represents the joint distribution of process states at times (Subscript[t, 1])<⋯<(Subscript[t, k]).}})

OpenersNotesSection(Spacer1)Details

(RefLink[SliceDistribution, paclet:ref/SliceDistribution])[proc,t] can be entered as proc[t].

(RefLink[SliceDistribution, paclet:ref/SliceDistribution])[proc,{(Subscript[t, 1]),…,(Subscript[t, k])}] can be entered as proc[{(Subscript[t, 1]),…,(Subscript[t, k])}].

For a random process x\[Distributed]proc, its state at time t is a random variable x[t]\[Distributed]proc[t], and its state at times (Subscript[t, 1]), …, (Subscript[t, k]) is a random variable {x[(Subscript[t, 1])],…,x[(Subscript[t, k])]}\[Distributed]proc[{(Subscript[t, 1]),…,(Subscript[t, k])}].

(RefLink[SliceDistribution, paclet:ref/SliceDistribution]) will simplify to known special distributions whenever possible.

(RefLink[SliceDistribution, paclet:ref/SliceDistribution]) can be used with such functions as (RefLink[Mean, paclet:ref/Mean]), (RefLink[CDF, paclet:ref/CDF]), and (RefLink[RandomVariate, paclet:ref/RandomVariate]), etc.

OpenersPrimaryExamplesSection(Spacer1)Examples  (8)

OpenersExampleSection0(Spacer1)Basic Examples  (3)

Find a univariate slice distribution of a (RefLink[PoissonProcess, paclet:ref/PoissonProcess]):

SliceDistribution[PoissonProcess[λ],t]

Find a bivariate slice distribution of a (RefLink[WienerProcess, paclet:ref/WienerProcess]):

WienerProcess[][{s,t}]

Find a multivariate slice distribution of a moving-average time series:

SliceDistribution[MAProcess[{2,4},1],{1,2,3}]

It does not autoevaluate but behaves like a distribution:

Mean[%]

OpenersExampleSection1(Spacer1)Scope  (3)

Slice distribution behaves like a distribution:

𝒟=SliceDistribution[TelegraphProcess[3],1]

Probability density function:

PDF[𝒟,x]

Characteristic function:

CharacteristicFunction[𝒟,x]

Moments:

Moment[𝒟,r]

Generate a set of pseudorandom numbers:

sample=RandomVariate[𝒟,20]

ListStepPlot[sample]

Slice distribution may autoevaluate to known distributions:

SliceDistribution[WienerProcess[μ,σ],t]

SliceDistribution[WienerProcess[μ,σ],{s,t}]

SliceDistribution[OrnsteinUhlenbeckProcess[μ,σ,θ],t]

SliceDistribution[WienerProcess[μ,σ],{s,t}]

SliceDistribution[OrnsteinUhlenbeckProcess[μ,σ,θ,(Subscript[x, 0])],t]

SliceDistribution[OrnsteinUhlenbeckProcess[μ,σ,θ,(Subscript[x, 0])],{s,t}]

SliceDistribution[BernoulliProcess[p],t]

SliceDistribution[BernoulliProcess[p],{s,t,w}]

SliceDistribution[BinomialProcess[p],t]

SliceDistribution[PoissonProcess[p],t]

SliceDistribution[CompoundPoissonProcess[λ,GammaDistribution[a,b]],t]

Slice distribution for an M/M/∞ queue:

𝒬=QueueingProcess[λ,μ,∞];

𝒟=SliceDistribution[𝒬,t];

Probability density function:

PDF[𝒟,x]

Sum[%,{x,0,∞}]

Cumulative distribution function:

CDF[𝒟,x]//FullSimplify

Mean of the slice distribution:

m=Mean[𝒟]

Find the limit of the mean as t approaches ∞:

Limit[m, t-> Infinity, Assumptions->λ>0&&μ>0]

This agrees with the mean of the corresponding (RefLink[StationaryDistribution, paclet:ref/StationaryDistribution]):

Mean[StationaryDistribution[𝒬]]

As well as the mean system size in the steady state:

QueueProperties[𝒬,"MeanSystemSize"]

OpenersExampleSection4(Spacer1)Properties & Relations  (2)

Slice distribution at infinity is (RefLink[StationaryDistribution, paclet:ref/StationaryDistribution]):

SliceDistribution[OrnsteinUhlenbeckProcess[μ,σ,θ,(Subscript[x, 0])],∞]

StationaryDistribution[OrnsteinUhlenbeckProcess[μ,σ,θ,(Subscript[x, 0])]]

Use implicit times for computing probabilities:

Probability[x[t]^2+x[t]<34,x\[Distributed]PoissonProcess[μ]]

Obtain the same result using the slice distribution:

Probability[x^2+x<34,x\[Distributed]PoissonProcess[μ][t]]

Compute an expectation using implicit time in the variable x[t]:

Expectation[x[t]^2+x[t]+3E^(-x[t]),x\[Distributed]PoissonProcess[μ]]

Obtain the same result using the slice distribution:

Expectation[x^2+x+3E^(-x),x\[Distributed]PoissonProcess[μ][t]]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[MarginalDistribution, paclet:ref/MarginalDistribution])  ▪  (RefLink[Distributed, paclet:ref/Distributed])  ▪  (RefLink[Probability, paclet:ref/Probability])  ▪  (RefLink[Expectation, paclet:ref/Expectation])  ▪  (RefLink[StationaryDistribution, paclet:ref/StationaryDistribution])  ▪  (RefLink[RandomFunction, paclet:ref/RandomFunction])  ▪  (RefLink[RandomVariate, paclet:ref/RandomVariate])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Random Processes, paclet:guide/RandomProcesses])}, {▪(RefLinkPlain[Stochastic Differential Equation Processes, paclet:guide/StochasticDifferentialEquationProcesses])}, {▪(RefLinkPlain[Derived Statistical Distributions, paclet:guide/DerivedDistributions])}, {▪(RefLinkPlain[Probability & Statistics, paclet:guide/ProbabilityAndStatistics])}})}})