Distribuciones Continuas Distribuciones Continuas
Distribuciones Discretas
ECUACIONES DISTRIBUCIÓN GAUSSIANA INVERSA 3P Definición de distribución X ∼ I n v e r s e G a u s s i a n 3 P ( μ , λ , Loc ) X\sim\mathrm{InverseGaussian_{3P}}\left(\mu,\lambda,\text{Loc}\right) X ∼ InverseGaussia n 3P ( μ , λ , Loc ) Dominio de distribución x ∈ ( 0 , ∞ ) x\in\left(0,\infty\right) x ∈ ( 0 , ∞ ) Dominio y restricciones de parámetros μ ∈ R + , λ ∈ R + , Loc ∈ R \mu\in\mathbb{R}^{+},\lambda\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R} μ ∈ R + , λ ∈ R + , Loc ∈ R Función de distribución acumulada F X ( x ) = Φ ( λ x − Loc ( x − Loc μ − 1 ) ) + exp ( 2 λ μ ) Φ ( − λ x − Loc ( x − Loc μ + 1 ) ) F_{X}\left(x\right)=\Phi\left(\sqrt{\frac{\lambda}{x-\text{Loc}}}\left(\frac{x-\text{Loc}}{\mu}-1\right)\right)+\exp\left(\frac{2 \lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x-\text{Loc}}}\left(\frac{x-\text{Loc}}{\mu}+1\right)\right) F X ( x ) = Φ ( x − Loc λ ( μ x − Loc − 1 ) ) + exp ( μ 2 λ ) Φ ( − x − Loc λ ( μ x − Loc + 1 ) ) Función de densidad de probabilidad f X ( x ) = λ 2 π ( x − Loc ) 3 exp [ − λ ( x − μ − Loc ) 2 2 μ 2 ( x − Loc ) ] f_{X}\left(x\right)=\sqrt\frac{\lambda}{2 \pi (x-\text{Loc})^3} \exp\left[-\frac{\lambda (x-\mu-\text{Loc})^2}{2 \mu^2 (x-\text{Loc})}\right] f X ( x ) = 2 π ( x − Loc ) 3 λ exp [ − 2 μ 2 ( x − Loc ) λ ( x − μ − Loc ) 2 ] Función de punto percentil Sample X = { Loc + x 0 if u 2 ⩽ μ μ + x 0 Loc + μ 2 x 0 if u 2 ⩾ μ μ + x 0 \text{Sample}_{X}=\left\{\begin{array}{cl} \text{Loc}+x_{0} \quad \text{if} \quad u_{2}\leqslant\frac{\mu}{\mu+x_{0}}\\ \text{Loc}+\frac{\mu^{2}}{x_{0}} \quad \text{if} \quad u_{2}\geqslant \frac{\mu}{\mu+x_{0}} \end{array} \right. Sample X = { Loc + x 0 if u 2 ⩽ μ + x 0 μ Loc + x 0 μ 2 if u 2 ⩾ μ + x 0 μ Momentos paramétricos no centrados μ k ′ = E [ X k ] = ∫ Loc ∞ x k f X ( x ) d x \mu'_{k}=E[X^k]=\int_{\text{Loc}}^{\infty }x^{k}f_{X}\left(x\right)dx μ k ′ = E [ X k ] = ∫ Loc ∞ x k f X ( x ) d x Media paramétrica M e a n ( X ) = μ 1 ′ = Loc + μ \mathrm{Mean}(X)=\mu'_{1}=\text{Loc}+\mu Mean ( X ) = μ 1 ′ = Loc + μ Varianza paramétrica V a r i a n c e ( X ) = μ 2 ′ − μ 1 ′ 2 = μ 3 λ \mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{\mu^3}{\lambda} Variance ( X ) = μ 2 ′ − μ 1 ′ 2 = λ μ 3 Coeficiente de asimetría paramétrico S k e w n e s s ( X ) = μ 3 ′ − 3 μ 2 ′ μ 1 ′ + 2 μ 1 ′ 3 ( μ 2 ′ − μ 1 ′ 2 ) 1.5 = 3 ( μ λ ) 1 / 2 \mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=3\left(\frac{\mu}{\lambda}\right)^{1/2} Skewness ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) 1.5 μ 3 ′ − 3 μ 2 ′ μ 1 ′ + 2 μ 1 ′ 3 = 3 ( λ μ ) 1/2 Curtosis paramétrica K u r t o s i s ( X ) = μ 4 ′ − 4 μ 1 ′ μ 3 ′ + 6 μ 1 ′ 2 μ 2 ′ − 3 μ 1 ′ 4 ( μ 2 ′ − μ 1 ′ 2 ) 2 = 3 + 15 μ λ \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}}=3+\frac{15 \mu}{\lambda} Kurtosis ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) 2 μ 4 ′ − 4 μ 1 ′ μ 3 ′ + 6 μ 1 ′ 2 μ 2 ′ − 3 μ 1 ′ 4 = 3 + λ 15 μ Mediana paramétrica M e d i a n ( X ) = F X − 1 ( 1 2 ) \mathrm{Median}(X)=F^{-1}_{X}\left(\frac{1}{2}\right) Median ( X ) = F X − 1 ( 2 1 ) Moda paramétrica M o d e ( X ) = Loc + μ [ ( 1 + 9 μ 2 4 λ 2 ) 1 2 − 3 μ 2 λ ] \mathrm{Mode}(X)=\text{Loc}+\mu\left[\left(1+\frac{9 \mu^2}{4 \lambda^2}\right)^\frac{1}{2}-\frac{3 \mu}{2 \lambda}\right] Mode ( X ) = Loc + μ [ ( 1 + 4 λ 2 9 μ 2 ) 2 1 − 2 λ 3 μ ] Información y definiciones adicionales Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution. \text{Computing an analytic expression for the inverse of the cumulative distribution function is not} \\ \text{feasible. Nonetheless, it is possible to generate a random sample from the distribution.} Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution. Loc : Location parameter \text{Loc}:\text{Location parameter} Loc : Location parameter Φ ( x ) : CDF normal standard distribution \Phi\left(x\right):\text{CDF normal standard distribution} Φ ( x ) : CDF normal standard distribution Φ − 1 ( x ) : PPF normal standard distribution \Phi^{-1}\left(x\right):\text{PPF normal standard distribution} Φ − 1 ( x ) : PPF normal standard distribution x 0 = μ + μ 2 [ Φ − 1 ( u 1 ) ] 2 2 λ − μ 2 λ 4 μ λ [ Φ − 1 ( u 1 ) ] 2 + μ 2 ( [ Φ − 1 ( u 1 ) ] 2 ) 2 x_{0}=\mu+\frac{\mu^2 [\Phi^{-1}\left(u_{1}\right)]^{2}}{2\lambda}-\frac{\mu}{2\lambda}\sqrt{4\mu \lambda [\Phi^{-1}\left(u_{1}\right)]^{2}+\mu^2 ([\Phi^{-1}\left(u_{1}\right)]^{2})^2} x 0 = μ + 2 λ μ 2 [ Φ − 1 ( u 1 ) ] 2 − 2 λ μ 4 μ λ [ Φ − 1 ( u 1 ) ] 2 + μ 2 ([ Φ − 1 ( u 1 ) ] 2 ) 2 u 1 : Uniform[0,1] random varible u_{1}:\text{Uniform[0,1] random varible} u 1 : Uniform[0,1] random varible u 2 : Uniform[0,1] random varible u_{2}:\text{Uniform[0,1] random varible} u 2 : Uniform[0,1] random varible