Weibull Distribution

Author

Eric W. Weisstein
February 16, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/WeibullDistribution.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Stats`

Plots

Show[GraphicsArray[{Plot[PDF[WeibullDistribution[2,1],x],{x,0,10},AxesLabel->TraditionalForm/@{x,P[x]},Ticks->None,DisplayFunction->Identity,PlotRange->All],Plot[CDF[WeibullDistribution[2,1],x],{x,0,10},AxesLabel->TraditionalForm/@{x,"\!\(\*
StyleBox[\"D\",\nFontSlant->\"Italic\"]\)(\!\(\*
StyleBox[\"x\",\nFontSlant->\"Italic\"]\))"},Ticks->None,DisplayFunction->Identity,PlotRange->All]}]]

Mathematica's form

Stats[WeibullDistribution[a,b],x]//StatsForm

PowerExpand[FullSimplify[Table[Integrate[x^n a b^(-a)Exp[-(x/b)^a]x^(-1+a),{x,0,Infinity},Assumptions->{Re[a]>0,
	a Abs[Arg[b]]<(π/2)}],{n,4}]]//TableForm]

({(Subscript[μ, 1]),(Subscript[μ, 2]),(Subscript[μ, 3]),(Subscript[μ, 4])}=PowerExpand[FullSimplify[Table[Integrate[(x-b Gamma[1+(1/a)])^n a b^(-a)Exp[-(x/b)^a]x^(-1+a),{x,0,Infinity},Assumptions->{Re[a]>0,
	a Abs[Arg[b]]<(π/2)}],{n,4}]]])//TableForm

Skewness

FullSimplify[PowerExpand[(Subscript[μ, 3])/((Subscript[μ, 2])^(3/2))]]

Kurtosis

FullSimplify[PowerExpand[(Subscript[μ, 4])/((Subscript[μ, 2])^2)-3]]

Characteristic Function

FourierTransform[a (b^(-a)) (E^(-((((x/b)))^a))) (x^(-1+a)) UnitStep[x],x,t]

Other form

(a/b) x^(a-1) Exp[-x^a/b]

PowerExpand[FullSimplify[Table[Table[Integrate[x^n (a/b) x^(a-1) Exp[-x^a/b],{x,0,Infinity},Assumptions->{Re[a]>0,Re[b]>0,x>0}]],{n,0,4}]]]//TableForm

({(Subscript[μ, 1]),(Subscript[μ, 2]),(Subscript[μ, 3]),(Subscript[μ, 4])}=PowerExpand[FullSimplify[Table[Table[Integrate[(x-((((1/b)))^(-1/a)) Gamma[1+(1/a)])^n (a/b) x^(a-1) Exp[-x^a/b],{x,0,Infinity},Assumptions->{Re[a]>0,Re[b]>0,x>0}]],{n,4}]]])//TableForm

Skewness

FullSimplify[(Subscript[μ, 3])/((Subscript[μ, 2])^(3/2))]

Kurtosis

FullSimplify[(Subscript[μ, 4])/((Subscript[μ, 2])^2)]