Core Algorithms

Compare Two Distributions with the Same Moment Sequence

Use varying scales to see fine-grained detail and general shapes of PDFs for two different distributions with equivalent moments, and compare their first several moments in tables.

In[1]:=

dist1 = ProbabilityDistribution[

1/4 Exp[-Abs[x]^(1/2)], {x, -Infinity, Infinity}];

dist2 = ProbabilityDistribution[

1/4 Exp[-Abs[x]^(1/2)] (1 + Cos[Sqrt[Abs[x]]]), {x, -Infinity,

Infinity}];

momentTable[dist_, kmax_, name_, color_] :=

Grid[Join[{{Style[name, Bold, FontSize -> 14,

FontFamily -> "Verdana"],

SpanFromLeft}, {Style["Order", Italic, FontFamily -> "Verdana"],

Style["Value", Italic, FontFamily -> "Verdana"]}},

Table[{Style[k, FontFamily -> "Verdana"],

Style[Moment[dist, k], FontFamily -> "Verdana"]}, {k, kmax}]],

Alignment -> {Left, Center},

Background -> {None, {{color, GrayLevel[.9]}}},

FrameStyle -> Directive[Thick, White],

Dividers -> {All, {White, {True}, White}}];

fgrid = Grid[{{TraditionalForm[

Moment[r] == HoldForm[(1 + (-1)^r)/2 (2 r + 1)!]]}},

Alignment -> {Left, Center},

Background -> {None, {{Hue[.6, .15, .9], GrayLevel[.9]}}},

FrameStyle -> Directive[Thick, White],

BaseStyle -> {Bold, FontSize -> 14, FontFamily -> "Verdana"},

Dividers -> {All, {White, {True}, White}}];

Row[{GraphicsColumn[

Table[LogPlot[{PDF[dist1, x], PDF[dist2, x]}, {x, -10^k, 10^k},

PlotRange -> All, Filling -> Axis], {k, 2, 4}],

ImageSize -> 300],

Column[{fgrid,

momentTable[dist1, 10,

TraditionalForm[Subscript[f, 1][x] == PDF[dist1, x]],

Lighter[ColorData[1][1], .6]],

momentTable[dist2, 10,

TraditionalForm[Subscript[f, 2][x] == PDF[dist2, x]],

Lighter[ColorData[1][2], .6]]}, Spacings -> 2]}]

Out[1]=