Distribuciones Continuas Distribuciones Continuas
Distribuciones Discretas
ECUACIONES DISTRIBUCIÓN CHI CUADRADO NO CENTRAL Definición de distribución X ∼ N o n C e n t r a l C h i S q u a r e ( λ , n ) X\sim\mathrm{NonCentralChiSquare}\left(\lambda,n\right) X ∼ NonCentralChiSquare ( λ , n ) Dominio de distribución x ∈ [ 0 , + ∞ ) x\in [0,+\infty) x ∈ [ 0 , + ∞ ) Dominio y restricciones de parámetros λ ∈ R + , n ∈ R + \lambda\in\mathbb{R}^{+},n\in\mathbb{R}^{+} λ ∈ R + , n ∈ R + Función de distribución acumulada F X ( x ) = 1 − Q n 2 ( λ , x ) F_{X}\left(x\right)=1-Q_{\frac{n}{2}}\left(\sqrt{\lambda},\sqrt{x}\right) F X ( x ) = 1 − Q 2 n ( λ , x ) Función de densidad de probabilidad f X ( x ) = 1 2 e − ( x + λ ) / 2 ( x λ ) n / 4 − 1 / 2 I n / 2 − 1 ( λ x ) f_{X}\left(x\right)=\frac{1}{2}e^{-(x+\lambda)/2} \left(\frac{x}{\lambda}\right)^{n/4-1/2}I_{n/2-1}(\sqrt{\lambda x}) f X ( x ) = 2 1 e − ( x + λ ) /2 ( λ x ) n /4 − 1/2 I n /2 − 1 ( λ x ) Función de punto percentil Sample X = ∑ i = 1 n ( λ n + Φ − 1 ( u i ) ) 2 \text{Sample}_{X}=\sum_{i=1}^{n}\left(\sqrt{\frac{\lambda}{n}}+\Phi^{-1}\left(u_{i}\right)\right)^{2} Sample X = ∑ i = 1 n ( n λ + Φ − 1 ( u i ) ) 2 Momentos paramétricos no centrados μ k ′ = E [ X k ] = ∫ 0 ∞ x k f X ( x ) d x = 2 k − 1 ( k − 1 ) ! ( n + k λ ) + ∑ j = 1 k − 1 ( k − 1 ) ! 2 j − 1 ( k − j ) ! ( n + j λ ) μ k − j ′ \mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{k}f_{X}\left(x\right)dx=2^{k-1}(k-1)!(n+k\lambda)+\sum_{j=1}^{k-1}\frac{(k-1)!2^{j-1}}{(k-j)!}(n+j\lambda )\mu'_{k-j} μ k ′ = E [ X k ] = ∫ 0 ∞ x k f X ( x ) d x = 2 k − 1 ( k − 1 )! ( n + kλ ) + ∑ j = 1 k − 1 ( k − j )! ( k − 1 )! 2 j − 1 ( n + jλ ) μ k − j ′ Media paramétrica M e a n ( X ) = μ 1 ′ = n + λ \mathrm{Mean}(X)=\mu'_{1}=n+\lambda Mean ( X ) = μ 1 ′ = n + λ Varianza paramétrica V a r i a n c e ( X ) = μ 2 ′ − μ 1 ′ 2 = 2 ( n + 2 λ ) \mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=2(n+2\lambda) Variance ( X ) = μ 2 ′ − μ 1 ′ 2 = 2 ( n + 2 λ ) 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 = 2 3 / 2 ( n + 3 λ ) ( n + 2 λ ) 3 / 2 \mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{2^{3/2}(n+3\lambda)}{(n+2\lambda)^{3/2}} Skewness ( X ) = ( μ 2 ′ − μ 1 ′ 2 ) 1.5 μ 3 ′ − 3 μ 2 ′ μ 1 ′ + 2 μ 1 ′ 3 = ( n + 2 λ ) 3/2 2 3/2 ( n + 3 λ ) 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 = 12 ( n + 4 λ ) ( n + 2 λ ) 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}}=\frac{12(n+4\lambda)}{(n+2\lambda)^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 = ( n + 2 λ ) 2 12 ( n + 4 λ ) 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 ) = 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. u i : Uniform[0,1] random varible u_{i}:\text{Uniform[0,1] random varible} u i : Uniform[0,1] random varible Φ − 1 ( x ) : PPF normal standard distribution \Phi^{-1}\left(x\right):\text{PPF normal standard distribution} Φ − 1 ( x ) : PPF normal standard distribution I α ( x ) : Modified Bessel function of the first kind of order α ∈ N I_{\alpha}\left(x\right):\text{Modified Bessel function of the first kind of order }\alpha\in\mathbb{N} I α ( x ) : Modified Bessel function of the first kind of order α ∈ N