Core Algorithms

Compute Binormal Probability over a Polygon

Probability of standard binormal variate with

over a regular octagon of unit radius expressed in terms of Owen's T function.

In[1]:=

In[2]:=

In[3]:=

In[4]:=

With[{sides = 8, r = 1/3},

Show[Plot3D[

PDF[BinormalDistribution[r], {x, y}], {x, -2, 2}, {y, -2, 2},

PlotStyle -> Opacity[0.5, Hue[.15, 0.2, .8]], Filling -> Axis,

FillingStyle -> {Black, Orange}, Lighting -> "Neutral",

BoundaryStyle -> None,

RegionFunction ->

PolygonRegionFunction[

Most[Table[{Cos[x], Sin[x]}, {x, 0, 2 \[Pi], (2 \[Pi])/sides}]]],

Mesh -> None, PlotRange -> All, NormalsFunction -> None,

PlotPoints -> 50],

Plot3D[PDF[BinormalDistribution[r], {x, y}], {x, -3, 3}, {y, -3, 3},

PlotStyle -> Opacity[0.2], Mesh -> 6, MeshStyle -> Gray,

PlotRange -> All, PlotPoints -> 50], Boxed -> False,

ImageSize -> 580,

Epilog ->

Inset[Framed[

Pane[Style[

Row[{TraditionalForm[Pr[1/3]] " \[Equal] ", 1,

FullSimplify[-4 OwenT[Sqrt[1/2 + 1/Sqrt[2]]/Sqrt[

Sqrt[2] - r], (Sqrt[2] - 1) Sqrt[(1 + r)/(1 - r)]] -

4 OwenT[Sqrt[1/2 + 1/Sqrt[2]]/Sqrt[Sqrt[2] - r], (

Sqrt[2] - 1 - r)/Sqrt[1 - r^2]] -

4 OwenT[Sqrt[1/2 + 1/Sqrt[2]]/Sqrt[

Sqrt[2] + r], (Sqrt[2] - 1) Sqrt[(1 - r)/(1 + r)]] -

4 OwenT[Sqrt[1/2 + 1/Sqrt[2]]/Sqrt[Sqrt[2] + r], (

Sqrt[2] - 1 + r)/Sqrt[1 - r^2]]]}], 11.5], {500, 118}],

Background -> LightYellow], {Center, Top}, {Center, Top}]]]

Out[4]=