PLAYGROUND

ECUACIONES DISTRIBUCIÓN SECANTE HIPERBÓLICA

Definición de distribución

X∼HyperbolicSecant(μ,σ)X\sim\mathrm{HyperbolicSecant}\left(\mu,\sigma\right)

Dominio de distribución

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Dominio y restricciones de parámetros

μ∈R,σ∈R+\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Función de distribución acumulada

FX(x)=2πarctan⁡[exp⁡ ⁣(π2 z(x))]F_{X}\left(x\right)=\frac{2}{\pi} \arctan\left[\exp\!\left(\frac{\pi}{2}\,z(x)\right)\right]

Función de densidad de probabilidad

fX(x)=12σsech⁡ ⁣(π2 z(x))f_{X}\left(x\right)=\frac{1}{2\sigma} \operatorname{sech}\!\left(\frac{\pi}{2}\,z(x)\right)

Función de punto percentil

FX−1(u)=μ+σ2π ln⁡ ⁣[tan⁡(π2 u)]F^{-1}_{X}\left(u\right)=\mu+\sigma\frac{2}{\pi}\,\ln\!\left[\tan\left(\frac{\pi}{2}\,u\right)\right]

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫−∞∞xkfX~(x)dx=1+(−1)k2π22kk![ζ(k+1,14)−ζ(k+1,34)]\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{-\infty}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{1+\left(-1\right)^{k}}{2\pi2^{2k}}k!\left[\zeta\left(k+1,\frac{1}{4}\right)-\zeta\left(k+1,\frac{3}{4}\right)\right]

Media paramétrica

Mean(X)=μ+σμ~1′=μ\mathrm{Mean}(X)=\mu+\sigma\tilde{\mu}'_{1}=\mu

Varianza paramétrica

Variance(X)=σ2(μ~2′−μ~1′2)=σ2\mathrm{Variance}(X)=\sigma^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\sigma^{2}

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=0\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}=0

Curtosis paramétrica

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}=3

Mediana paramétrica

Median(X)=μ\mathrm{Median}(X)=\mu

Moda paramétrica

Mode(X)=μ\mathrm{Mode}(X)=\mu

Información y definiciones adicionales

X~∼HyperbolicSecant(0,1)\tilde{X}\sim\mathrm{HyperbolicSecant}\left(0,1\right)
μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
z(x)=(x−μ)/σz\left(x\right)=\left(x-\mu\right)/\sigma
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
ζ(a,s):Hurwitz zeta function\zeta(a,s):\text{Hurwitz zeta function}