PLAYGROUND

ECUACIONES DISTRIBUCIÓN ARGUS

Definición de distribución

X∼Argus(χ,Loc,Sc)X\sim\mathrm{Argus}\left(\chi,\text{Loc},\text{Sc}\right)

Dominio de distribución

x∈(Loc,Loc+Sc)x\in\left(\text{Loc},\text{Loc}+\text{Sc}\right)

Dominio y restricciones de parámetros

χ∈R+,Loc∈R,Sc∈R+\chi\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}

Función de distribución acumulada

FX(x)=1−Ψ(χ1−z(x)2)Ψ(χ)F_{X}\left(x\right)=1-\frac{\Psi\left(\chi\sqrt{1-z(x)^2}\right)}{\Psi(\chi)}

Función de densidad de probabilidad

fX(x)=1Sc⋅χ32π Ψ(χ)⋅z(x)1−z(x)2exp⁡(−12χ2(1−z(x)2))f_{X}\left(x\right)=\frac{1}{\text{Sc}}\cdot\frac{\chi^3}{\sqrt{2\pi}\,\Psi(\chi)}\cdot z(x)\sqrt{1-z(x)^2}\exp\left(-\frac12 \chi^2\Big(1-z(x)^2\Big)\right)

Función de punto percentil

FX−1(u)=Loc+Sc1−2P−1(32,(1−u)P(32,χ22))χ2F^{-1}_{X}\left(u\right)=\text{Loc}+\text{Sc}\sqrt{1-\frac{2\text{P}^{-1}(\frac{3}{2},(1-u)\text{P}(\frac{3}{2},\frac{\chi^{2}}{2}))}{\chi^{2}}}

Momentos paramétricos no centrados

μk′=E[Xk]=∫LocLoc+ScxkfX(x)dx\mu'_{k}=E[X^k]=\int_{\text{Loc}}^{\text{Loc}+\text{Sc}}x^{k}f_{X}\left(x\right)dx

Media paramétrica

Mean(X)=μ1′=Loc+Scπ/8 χe−χ24I1(χ24)Ψ(χ)\mathrm{Mean}(X)=\mu'_{1}=\text{Loc}+\text{Sc}\sqrt{\pi/8}\,\frac{\chi e^{-\frac{\chi^2}{4}} I_1(\tfrac{\chi^2}{4})}{\Psi(\chi)}

Varianza paramétrica

Variance(X)=μ2′−μ1′2=Sc2⋅(1−3χ2+χϕ(χ)Ψ(χ))−(μ−Loc)2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\text{Sc}^2\cdot\left(1-\frac{3}{\chi^2}+\frac{\chi\phi(\chi)}{\Psi(\chi)}\right)-(\mu-\text{Loc})^2

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Curtosis paramétrica

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Mediana paramétrica

Median(X)=Loc+Sc1−2P−1(32,12P(32,χ22))χ2\mathrm{Median}(X)=\text{Loc}+\text{Sc}\sqrt{1-\frac{2\text{P}^{-1}(\frac{3}{2},\frac{1}{2}\text{P}(\frac{3}{2},\frac{\chi^{2}}{2}))}{\chi^{2}}}

Moda paramétrica

Mode(X)=Loc+Sc2χ(χ2−2)+χ4+4\mathrm{Mode}(X)=\text{Loc}+\frac{\text{Sc}}{\sqrt2\chi}\sqrt{(\chi^2-2)+\sqrt{\chi^4+4}}

Información y definiciones adicionales

Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz(x)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Ψ(χ)=Φ(χ)−χϕ(χ)−12\Psi(\chi)=\Phi(\chi)- \chi \phi( \chi )-\tfrac{1}{2}
Φ(x):CDF normal standard distribution\Phi(x):\text{CDF normal standard distribution}
ϕ(x):PDF normal standard distribution\phi(x):\text{PDF normal standard distribution}
Iα(x):Modified Bessel function of the first kind of order α∈NI_{\alpha}\left(x\right):\text{Modified Bessel function of the first kind of order }\alpha\in\mathbb{N}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}(a,x)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
P−1(a,y):Inverse of regularized lower incomplete gamma function\text{P}^{-1}(a,y):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma(a,x):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma(x):\text{Gamma function}