Distribuciones Continuas Distribuciones Continuas
Distribuciones Discretas
ECUACIONES DISTRIBUCIÓN NORMAL PLEGADA Definición de distribución X ∼ F o l d e d N o r m a l ( μ , σ ) X\sim\mathrm{FoldedNormal}\left(\mu,\sigma\right) X ∼ FoldedNormal ( μ , σ ) Dominio de distribución x ∈ [ 0 , ∞ ) x\in [0,\infty) x ∈ [ 0 , ∞ ) Dominio y restricciones de parámetros μ ∈ R , σ ∈ R + \mu\in\mathbb{R},\sigma\in\mathbb{R}^{+} μ ∈ R , σ ∈ R + Función de distribución acumulada F X ( x ) = 1 2 [ erf ( x + μ σ 2 ) + erf ( x − μ σ 2 ) ] F_{X}\left(x\right)=\frac{1}{2}\left[\text{erf}\left(\frac{x+\mu}{\sigma\sqrt{2}}\right)+\text{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right] F X ( x ) = 2 1 [ erf ( σ 2 x + μ ) + erf ( σ 2 x − μ ) ] Función de densidad de probabilidad f X ( x ) = 1 σ 2 π e − ( x − μ ) 2 2 σ 2 + 1 σ 2 π e − ( x + μ ) 2 2 σ 2 f_{X}\left(x\right)=\frac{1}{\sigma\sqrt{2\pi}} \,e^{ -\frac{(x-\mu)^2}{2\sigma^2} }+\frac{1}{\sigma\sqrt{2\pi}} \,e^{ -\frac{(x+\mu)^2}{2\sigma^2} } f X ( x ) = σ 2 π 1 e − 2 σ 2 ( x − μ ) 2 + σ 2 π 1 e − 2 σ 2 ( x + μ ) 2 Función de punto percentil Sample X ( u ) = ∣ μ + σ Φ − 1 ( u ) ∣ \text{Sample}_{X}\left(u\right)=\left|\mu+\sigma\Phi^{-1}(u)\right| Sample X ( u ) = μ + σ Φ − 1 ( u ) Momentos paramétricos no centrados μ k ′ = E [ X k ] = ∫ 0 ∞ x k f X ( x ) d x \mu'_{k}=E[X^k]=\int_{0}^{\infty }x^{k}f_{X}\left(x\right)dx μ k ′ = E [ X k ] = ∫ 0 ∞ x k f X ( x ) d x Media paramétrica M e a n ( X ) = μ 1 ′ = σ 2 π e ( − μ 2 / 2 σ 2 ) + μ ( 1 − 2 Φ ( − μ σ ) ) \mathrm{Mean}(X)=\mu'_{1}=\sigma \sqrt{\tfrac{2}{\pi}} \,e^{(-\mu^2/2\sigma^2)}+\mu\left(1-2\,\Phi(-\tfrac{\mu}{\sigma})\right) Mean ( X ) = μ 1 ′ = σ π 2 e ( − μ 2 /2 σ 2 ) + μ ( 1 − 2 Φ ( − σ μ ) ) Varianza paramétrica V a r i a n c e ( X ) = μ 2 ′ − μ 1 ′ 2 = μ 2 + σ 2 − M e a n ( X ) 2 \mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\mu^2+\sigma^2-\mathrm{Mean}(X)^{2} Variance ( X ) = μ 2 ′ − μ 1 ′ 2 = μ 2 + σ 2 − Mean ( X ) 2 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 \mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}} Skewness ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) 1.5 μ 3 ′ − 3 μ 2 ′ μ 1 ′ + 2 μ 1 ′ 3 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 \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}} 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 Mediana paramétrica M e d i a n ( X ) = ∣ μ + σ Φ − 1 ( 1 / 2 ) ∣ \mathrm{Median}(X)=\left|\mu+\sigma\Phi^{-1}\left(1/2\right)\right| Median ( X ) = μ + σ Φ − 1 ( 1/2 ) Moda paramétrica M o d e ( X ) = arg max x f X ( x ) \mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right) Mode ( X ) = arg max x f X ( x ) 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. μ : Location parameter \mu:\text{Location parameter} μ : Location parameter σ : Scale parameter \sigma:\text{Scale parameter} σ : Scale parameter u : Uniform[0,1] random varible u:\text{Uniform[0,1] random varible} u : Uniform[0,1] random varible Φ ( x ) : CDF normal standard distribution \Phi\left(x\right):\text{CDF normal standard distribution} Φ ( x ) : CDF normal standard distribution ϕ ( x ) : PDF normal standard distribution \phi\left(x\right):\text{PDF normal standard distribution} ϕ ( x ) : PDF 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