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SliceDistribution
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SliceDistribution

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represents the distribution of the process state at time t.

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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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Find a univariate slice distribution of a PoissonProcess:

In[1]:=1
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https://wolfram.com/xid/0c0osliun7nve-dh1n61
Out[1]=1

Find a bivariate slice distribution of a WienerProcess:

In[1]:=1
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https://wolfram.com/xid/0c0osliun7nve-xogvga
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0c0osliun7nve-4it02
Out[1]=1

It does not autoevaluate but behaves like a distribution:

In[2]:=2
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https://wolfram.com/xid/0c0osliun7nve-cjrcx7
Out[2]=2

Scope  (3)Survey of the scope of standard use cases

Slice distribution behaves like a distribution:

In[1]:=1
✖
https://wolfram.com/xid/0c0osliun7nve-d7dr3y
Out[1]=1

Probability density function:

In[2]:=2
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https://wolfram.com/xid/0c0osliun7nve-l8ustk
Out[2]=2

Characteristic function:

In[3]:=3
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https://wolfram.com/xid/0c0osliun7nve-ye0ir5
Out[3]=3

Moments:

In[4]:=4
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https://wolfram.com/xid/0c0osliun7nve-jd6hzl
Out[4]=4

Generate a set of pseudorandom numbers:

In[5]:=5
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https://wolfram.com/xid/0c0osliun7nve-4u8npo
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0c0osliun7nve-kxs8mv
Out[6]=6

Slice distribution may autoevaluate to known distributions:

In[1]:=1
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https://wolfram.com/xid/0c0osliun7nve-7spis
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c0osliun7nve-h4pb4n
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0c0osliun7nve-8ddxlr
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0c0osliun7nve-g81x1v
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0c0osliun7nve-4m9sc3
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0c0osliun7nve-v8h92t
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0c0osliun7nve-5n3nir
Out[7]=7
In[8]:=8
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https://wolfram.com/xid/0c0osliun7nve-74llxu
Out[8]=8
In[9]:=9
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https://wolfram.com/xid/0c0osliun7nve-htrfso
Out[9]=9
In[10]:=10
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https://wolfram.com/xid/0c0osliun7nve-0khrrk
Out[10]=10
In[11]:=11
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https://wolfram.com/xid/0c0osliun7nve-42pfim
Out[11]=11

Slice distribution for an M/M/ queue:

In[1]:=1
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https://wolfram.com/xid/0c0osliun7nve-xpm09
In[2]:=2
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https://wolfram.com/xid/0c0osliun7nve-mcrvhu

Probability density function:

In[3]:=3
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https://wolfram.com/xid/0c0osliun7nve-g7aiel
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0c0osliun7nve-dph6fm
Out[4]=4

Cumulative distribution function:

In[5]:=5
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https://wolfram.com/xid/0c0osliun7nve-gud1e
Out[5]=5

Mean of the slice distribution:

In[6]:=6
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https://wolfram.com/xid/0c0osliun7nve-ghwos3
Out[6]=6

Find the limit of the mean as t approaches :

In[7]:=7
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https://wolfram.com/xid/0c0osliun7nve-e0ha7c
Out[7]=7

This agrees with the mean of the corresponding StationaryDistribution:

In[8]:=8
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https://wolfram.com/xid/0c0osliun7nve-b018gm
Out[8]=8

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

In[9]:=9
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https://wolfram.com/xid/0c0osliun7nve-hevbgs
Out[9]=9

Properties & Relations  (2)Properties of the function, and connections to other functions

Slice distribution at infinity is StationaryDistribution:

In[1]:=1
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https://wolfram.com/xid/0c0osliun7nve-nvlxxm
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c0osliun7nve-0olw08
Out[2]=2

Use implicit times for computing probabilities:

In[1]:=1
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https://wolfram.com/xid/0c0osliun7nve-ml0csm
Out[1]=1

Obtain the same result using the slice distribution:

In[2]:=2
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https://wolfram.com/xid/0c0osliun7nve-casj6p
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0c0osliun7nve-drrgim
Out[3]=3

Obtain the same result using the slice distribution:

In[4]:=4
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https://wolfram.com/xid/0c0osliun7nve-fdue9q
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
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https://wolfram.com/xid/0c0osliun7nve-rfs9le
In[23]:=23
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https://wolfram.com/xid/0c0osliun7nve-kdqr36
Out[23]=23

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

In[3]:=3
✖
https://wolfram.com/xid/0c0osliun7nve-us1z3r
Out[3]=3

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

In[19]:=19
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https://wolfram.com/xid/0c0osliun7nve-evl5nv
In[22]:=22
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https://wolfram.com/xid/0c0osliun7nve-vskkdp
Out[22]=22
Wolfram Research (2012), SliceDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/SliceDistribution.html.
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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.

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_slicedistribution, author="Wolfram Research", title="{SliceDistribution}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SliceDistribution.html}", note=[Accessed: 29-April-2025 ]}

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@misc{reference.wolfram_2025_slicedistribution, author="Wolfram Research", title="{SliceDistribution}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SliceDistribution.html}", note=[Accessed: 29-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_slicedistribution, organization={Wolfram Research}, title={SliceDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/SliceDistribution.html}, note=[Accessed: 29-April-2025 ]}

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@online{reference.wolfram_2025_slicedistribution, organization={Wolfram Research}, title={SliceDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/SliceDistribution.html}, note=[Accessed: 29-April-2025 ]}