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Random Processes
Slice Distribution for Processes
A random process
at time
is a random variable
given by its
SliceDistribution
. The slice distribution is plotted for a range of time values.
A discretetime and discretestate random process.
In[1]:=
X
SliceDistribution[BinomialProcess[2/3], t]
Out[1]=
In[2]:=
X
SliceDistribution[BinomialProcess[2/3], t]; DiscretePlot3D[PDF[%, x], {t, 0, 10}, {x, 0, 10}, ExtentSize > 1/2, PlotStyle > EdgeForm[Opacity[0.3]], FillingStyle > Opacity[0.4], AxesLabel > Automatic, ColorFunction > "Rainbow", ImageSize > Medium]
Out[2]=
A discretetime and continuousstate random process.
In[3]:=
X
SliceDistribution[ARProcess[{2/3}, 1], t]
Out[3]=
In[4]:=
X
SliceDistribution[ARProcess[{2/3}, 1], t]; Plot3D[Evaluate[ PDF[%, x] Sum[UnitBox[2 (t  i)], {i, 0, 10}]], {t, 0.5, 10.5}, {x, 5, 5}, MeshFunctions > {#1 &}, Mesh > {Sort[Join[Range[0, 10]  0.25, Range[0, 10] + 0.25]]}, MeshShading > {None, Automatic}, Exclusions > {Sin[\[Pi] (t  0.25)], Sin[\[Pi] (t + 0.25)]}, AxesLabel > Automatic, BoundaryStyle > None, ColorFunction > "Rainbow", ImageSize > Medium]
Out[4]=
A continuoustime and discretestate random process.
In[5]:=
X
SliceDistribution[PoissonProcess[1], t]
Out[5]=
In[6]:=
X
SliceDistribution[PoissonProcess[1], t]; Plot3D[Evaluate[ PDF[%, Floor[x  0.25]] Sum[UnitBox[2 (x  i)], {i, 0, 10}]], {t, 1, 10}, {x, 0.5, 10.5}, MeshFunctions > {#2 &}, Mesh > {Sort[Join[Range[0, 10]  0.25, Range[0, 10] + 0.25]]}, MeshShading > {None, Automatic}, Exclusions > {Sin[\[Pi] (x  0.25)], Sin[\[Pi] (x + 0.25)]}, AxesLabel > Automatic, PlotRange > All, PlotPoints > 75, BoundaryStyle > None, Ticks > {Automatic, Range[0, 10, 5], Automatic}, ColorFunction > "Rainbow", ImageSize > Medium]
Out[6]=
A continuoustime and continuousstate random process.
In[7]:=
X
SliceDistribution[WienerProcess[1, 1], t]
Out[7]=
In[8]:=
X
SliceDistribution[WienerProcess[1, 1], t]; Plot3D[PDF[%, x], {t, 1, 10}, {x, 3 Sqrt[10] + 5, 3 Sqrt[10] + 5}, AxesLabel > Automatic, PlotRange > All, MeshFunctions > {#1 &}, Mesh > {Range[9]}, PlotPoints > 50, ColorFunction > "Rainbow", ImageSize > Medium]
Out[8]=