Gamma Distribution

Author

Eric W. Weisstein
August 20, 2008

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

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

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

Plots

V5

Show[GraphicsArray[Block[{$DisplayFunction=Identity},Function[f,Plot[Evaluate[f[GammaDistribution[#,1],x]&/@Range[3]],{x,.01,10},PlotStyle->{Black,Red,Blue},AxesLabel->TraditionalForm/@{x,f[x]}]]/@{PDF,CDF}
]],ImageSize->500]

V6

GraphicsRow[Function[f,Plot[Evaluate[f[GammaDistribution[#,1],x]&/@Range[3]],{x,.01,10},PlotStyle->{Black,Red,Blue},AxesLabel->TraditionalForm/@{x,Row[{Style[f,Italic]/.{CDF->"D",PDF->"P"},"(",x,")"}]}]]/@{PDF,CDF}]

Stats

Stats[GammaDistribution[α,θ],x]//StatsForm//TraditionalForm

PDF

v5

Integrate[PDF[GammaDistribution[α,θ],x],{x,0,∞}]

v6

Integrate[PDF[GammaDistribution[α,θ],x],{x,0,∞}]

Integrate[PDF[GammaDistribution[α,θ],x],{x,0,∞},Assumptions->θ>0&&α>0]

CDF[GammaDistribution[α,θ],x]

Moments

Integrate[x^n PDF[GammaDistribution[α,θ],x],{x,0,∞},Assumptions->θ>0&&α>0&&n≥0]

FullSimplify[Table[(((θ^n) Gamma[n+α])/(Gamma[α])),{n,0,3}]]

Table[Integrate[x^n PDF[GammaDistribution[α,θ],x],{x,0,∞},Assumptions->θ>0&&α>0],{n,0,3}]//FullSimplify

Characteristic Function

v5.1

FullSimplify[FourierTransform[(((E^(-x/θ)) (x^(-1+α)) (θ^(-α))  UnitStep[x])/(Gamma[α])),x,t],α>0&&θ>0]

FullSimplify[%,t>0&&α>0&&θ>0]

v6

FullSimplify[FourierTransform[(((E^(-x/θ)) (x^(-1+α)) (θ^(-α))  UnitStep[x])/(Gamma[α])),x,t],α>0&&θ>0]

FullSimplify[%,t>0&&α>0&&θ>0]

Transformations

Factor//@Solve[v==((x1/(θ1))/(((x2 )/(θ2))+(x1/(θ1))))&&u==(x1/(θ1))+(x2/(θ2)),{x1,x2}]

With[{v=((x1/(θ1))/(((x2 )/(θ2))+(x1/(θ1)))),u=(x1/(θ1))+(x2/(θ2))},
Det[{{v,u},{1-v,-u}}]/u
]//Simplify

FullSimplify[((Exp[-u](((u v (Subscript[θ, 1])))^((Subscript[α, 1])-1))(((u(1-v)(Subscript[θ, 2])))^((Subscript[α, 2])-1))u)/(Gamma[(Subscript[α, 1])]Gamma[(Subscript[α, 2])]((Subscript[θ, 1])^(Subscript[α, 1]))((Subscript[θ, 2])^(Subscript[α, 2])))),(Subscript[α, 1])>1&&(Subscript[α, 2])>2&&(Subscript[θ, 1])>0&&(Subscript[θ, 2])>0&&u>0&&v>0]

Integrate[((Exp[-u](u^((Subscript[α, 1])+(Subscript[α, 2])-1))(v^((Subscript[α, 1])-1))(((1-v))^((Subscript[α, 2])-1)))/(Gamma[(Subscript[α, 1])]Gamma[(Subscript[α, 2])](Subscript[θ, 1])(Subscript[θ, 2]))),{v,0,1},Assumptions->(Subscript[α, 1])>0&&(Subscript[α, 2])>0]

Sums

1-(∑_(k=0)^(h-1))((exp(x (-λ)) (((x λ))^k))/(k!))==1-((Γ(h,x λ))/(Γ(h)))

1-Sum[(((((λ x))^k)Exp[-λ x])/(k!)),{k,0,h-1}]

FullSimplify[%]

λ exp(x (-λ)) (∑_(k=0)^(h-1))((((x λ))^k)/(k!))-exp(x (-λ))(∑_(k=0)^(h-1))((k λ (((x λ))^(k-1)))/(k!))==(((x^(h-1)) (λ^h) (e^(-x λ)))/(Γ(h)))

λ Exp[-λ x]Sum[((((λ x))^k)/(k!)),{k,0,h-1}]-Exp[-λ x]Sum[((k(((λ x))^(k-1))λ)/(k!)),{k,0,h-1}]

FullSimplify[%,x>0]