# SliceDistribution

SliceDistribution[proc,t]

represents the distribution of the process state at time t.

SliceDistribution[proc,{t1,,tk}]

represents the joint distribution of process states at times t1<<tk.

# Details

• SliceDistribution[proc,t] can be entered as proc[t].
• SliceDistribution[proc,{t1,,tk}] can be entered as proc[{t1,,tk}].
• For a random process xproc, its state at time t is a random variable x[t]proc[t], and its state at times t1, , tk is a random variable {x[t1],,x[tk]}proc[{t1,,tk}].
• SliceDistribution will simplify to known special distributions whenever possible.
• SliceDistribution can be used with such functions as Mean, CDF, and RandomVariate, etc.

# Examples

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## Basic Examples(3)

Find a univariate slice distribution of a PoissonProcess:

Find a bivariate slice distribution of a WienerProcess:

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

It does not autoevaluate but behaves like a distribution:

## Scope(3)

Slice distribution behaves like a distribution:

Probability density function:

Characteristic function:

Moments:

Generate a set of pseudorandom numbers:

Slice distribution may autoevaluate to known distributions:

Slice distribution for an M/M/ queue:

Probability density function:

Cumulative distribution function:

Mean of the slice distribution:

Find the limit of the mean as t approaches :

This agrees with the mean of the corresponding StationaryDistribution:

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

## Properties & Relations(2)

Slice distribution at infinity is StationaryDistribution:

Use implicit times for computing probabilities:

Obtain the same result using the slice distribution:

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

Obtain the same result using the slice distribution:

## Possible Issues(1)

For some continuous-time random processes, simulation of a slice distribution is not well defined:

The process path simulation between the origin and the end time depends on the choice of step:

The slice distribution simulations for a few step choices show the approximations of the exact slice distribution:

Wolfram Research (2012), SliceDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SliceDistribution.html.

#### Text

Wolfram Research (2012), SliceDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SliceDistribution.html.

#### CMS

Wolfram Language. 2012. "SliceDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SliceDistribution.html.

#### APA

Wolfram Language. (2012). SliceDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SliceDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_slicedistribution, author="Wolfram Research", title="{SliceDistribution}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SliceDistribution.html}", note=[Accessed: 07-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_slicedistribution, organization={Wolfram Research}, title={SliceDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/SliceDistribution.html}, note=[Accessed: 07-September-2024 ]}