Map-Airy Distribution

Author

Eric W. Weisstein
November 23, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/Map-AiryDistribution.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Map-AiryDistribution.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

MapAiryDistribution /: PDF[MapAiryDistribution,x_]:=2 Exp[-2/3 x^3](x AiryAi[x^2]-AiryAiPrime[x^2])
MapAiryDistribution /:CDF[MapAiryDistribution,x_]:=1/3+(2*x*HypergeometricPFQ[{-(1/6),1/3},{-(1/3),4/3},-((4*x^3)/3)])/(3^(1/3)*Gamma[1/3])+(x^2*HypergeometricPFQ[{1/6,2/3},{1/3,5/3},-((4*x^3)/3)])/(3^(2/3)*Gamma[2/3])-(x^4*HypergeometricPFQ[{5/6,4/3},{5/3,7/3},-((4*x^3)/3)])/(6*3^(1/3)*Gamma[4/3])-(2*x^5*HypergeometricPFQ[{7/6,5/3},{7/3,8/3},-((4*x^3)/3)])/(15*3^(2/3)*Gamma[5/3])
MapAiryDistribution /:Domain[MapAiryDistribution]:=Interval[{-Infinity,Infinity}]
MapAiryDistribution /:Mean[MapAiryDistribution]:=0

Plot

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
Plot[#[MapAiryDistribution,x],{x,-2.5,2.5},
PlotRange->All,
AxesLabel->TraditionalForm/@{x,#[x]},
PlotStyle->Red]&/@{PDF,CDF}
]
],GraphicsSpacing->-.07,ImageSize->500]//Timing

Series

V5

<<DiscreteMath`RSolve`

SeriesTerm[PDF[MapAiryDistribution,x],{x,0,n}]

V7

SeriesCoefficient[PDF[MapAiryDistribution,x],{x,0,n}]//Timing

Series

s=Series[PDF[MapAiryDistribution,x],{x,0,20}]

<<Utilities`Simplify`

GammaSimplify[s]

t=Transpose[Partition[CoefficientList[s,x],3]];

t[[1]]Gamma[1/3]3^(1/3)

t[[2]]Gamma[2/3]3^(2/3)

Limits

v5.2

Limit[PDF[MapAiryDistribution,x],x->∞#]&/@{-1,1}

Limit[CDF[MapAiryDistribution,x],x->∞#]&/@{-1,1}

v6

Limit[PDF[MapAiryDistribution,x],x->∞#]&/@{-1,1}

Limit[CDF[MapAiryDistribution,x],x->∞#]&/@{-1,1}//FullSimplify

CDF[MapAiryDistribution,x]/.x->-N[4,50]

v7

Limit[PDF[MapAiryDistribution,x],x->∞#]&/@{-1,1}

Limit[CDF[MapAiryDistribution,x],x->∞#]&/@{-1,1}//FullSimplify

cdf=CDF[MapAiryDistribution,x]

Limit[cdf,x->-∞]

CDF[MapAiryDistribution,-10.`600]

CDF

FullSimplify[D[CDF[MapAiryDistribution,x],x]]

%-PDF[MapAiryDistribution,x]

Normalization

Correct answer is 1

Numerical

NIntegrate[PDF[MapAiryDistribution,x],{x,-100,100}]//Timing

V4.2

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity},GenerateConditions->False]//Timing

NIntegrate[PDF[MapAiryDistribution,x],{x,-100,100}]//Timing

NIntegrate[PDF[MapAiryDistribution,x],{x,-100,-2,2,100}]//Timing

V5.0

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity},GenerateConditions->False]//Timing

V5.2

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity},GenerateConditions->False]//Timing

V6

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity},GenerateConditions->False]//Timing

V7

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[PDF[MapAiryDistribution,x],{x,-Infinity,Infinity},GenerateConditions->False]//Timing

Mean

Correct answer is 0

v4.2

Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

v5.0

Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,Infinity},GenerateConditions->False]//Timing

v5.2

Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,Infinity},GenerateConditions->False]//Timing

FullSimplify[Integrate[x PDF[MapAiryDistribution,x],{x,0,Infinity}]+Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,0}]]

v6

Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

V7

Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

NIntegrate[x PDF[MapAiryDistribution,x],{x,-100,100}]//Timing

Two Halfs of Mean

V5.2

FullSimplify[{Integrate[x PDF[MapAiryDistribution,x],{x,0,Infinity}],Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,0}]}]//Timing

v6

FullSimplify[{Integrate[x PDF[MapAiryDistribution,x],{x,0,Infinity}],Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,0}]}]//Timing

Development V6:

FullSimplify[{Integrate[x PDF[MapAiryDistribution,x],{x,0,Infinity}],Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,0}]}]//Timing

<<Utilities`Simplify`

FullSimplify[GammaSimplify[%13]]

V7

FullSimplify[{Integrate[x PDF[MapAiryDistribution,x],{x,0,Infinity}],Integrate[x PDF[MapAiryDistribution,x],{x,-Infinity,0}]}]//Timing

Central Moments

2nd

v4.2

Integrate[x^2 PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[x^3 PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

v5.0

Integrate[x^2 PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

v5.2

Integrate[x^2 PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[x^2 PDF[MapAiryDistribution,x],{x,-Infinity,0}]+Integrate[x^2 PDF[MapAiryDistribution,x],{x,0,Infinity}]//Timing//Timing

v6

{Integrate[x^2 PDF[MapAiryDistribution,x],{x,-Infinity,0}],Integrate[x^2 PDF[MapAiryDistribution,x],{x,0,Infinity}]}//Timing

NIntegrate[x^2 PDF[MapAiryDistribution,x],{x,0,∞}]

N[-((9 (3^(2/3)))/(80 Gamma[-(8/3)]))]

NIntegrate[x^2 PDF[MapAiryDistribution,x],{x,-Infinity,0},WorkingPrecision->20,MaxRecursion->50]//Timing

3nd Central Moment

v5.2

Integrate[x^3 PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

Integrate[x^3 PDF[MapAiryDistribution,x],{x,-Infinity,0}]+Integrate[x^3 PDF[MapAiryDistribution,x],{x,0,Infinity}]//Timing//Timing

v6

Integrate[x^3 PDF[MapAiryDistribution,x],{x,-Infinity,Infinity}]//Timing

{Integrate[x^3 PDF[MapAiryDistribution,x],{x,-Infinity,0}],Integrate[x^3 PDF[MapAiryDistribution,x],{x,0,Infinity}]}//Timing

4th Central Moment

v6

{Integrate[x^4 PDF[MapAiryDistribution,x],{x,-Infinity,0}],Integrate[x^4 PDF[MapAiryDistribution,x],{x,0,Infinity}]}//Timing