Distribuciones Continuas Distribuciones Continuas
Distribuciones Discretas
ECUACIONES DISTRIBUCIÓN F NO CENTRAL Definición de distribución X ∼ N o n C e n t r a l F ( λ , n 1 , n 2 ) X\sim\mathrm{NonCentralF}\left(\lambda,n_{1},n_{2}\right) X ∼ NonCentralF ( λ , n 1 , n 2 ) Dominio de distribución x ∈ [ 0 , ∞ ) x\in [0,\infty) x ∈ [ 0 , ∞ ) Dominio y restricciones de parámetros λ ∈ R + , n 1 ∈ R + , n 2 ∈ R + \lambda\in\mathbb{R}^{+},n_{1}\in\mathbb{R}^{+},n_{2}\in\mathbb{R}^{+} λ ∈ R + , n 1 ∈ R + , n 2 ∈ R + Función de distribución acumulada F X ( x ) = ∑ j = 0 ∞ ( ( 1 2 λ ) j j ! e − λ / 2 ) I n 1 x / ( n 2 + n 1 x ) ( n 1 2 + j , n 2 2 ) F_{X}\left(x\right)=\sum\limits_{j=0}^\infty\left(\frac{\left(\frac{1}{2}\lambda\right)^j}{j!}e^{-\lambda/2}\right)I_{n_1x/(n_2+n_1x)}\left(\frac{n_1}{2}+j,\frac{n_2}{2}\right) F X ( x ) = j = 0 ∑ ∞ ( j ! ( 2 1 λ ) j e − λ /2 ) I n 1 x / ( n 2 + n 1 x ) ( 2 n 1 + j , 2 n 2 ) Función de densidad de probabilidad f X ( x ) = ∑ k = 0 ∞ e − λ / 2 ( λ / 2 ) k Beta ( n 2 2 , n 1 2 + k ) k ! ( n 1 n 2 ) n 1 2 + k ( n 2 n 2 + n 1 x ) n 1 + n 2 2 + k x n 1 / 2 − 1 + k f_{X}\left(x\right)=\sum\limits_{k=0}^\infty\frac{e^{-\lambda/2}(\lambda/2)^k}{ \text{Beta}\left(\frac{n_2}{2},\frac{n_1}{2}+k\right) k!}\left(\frac{n_1}{n_2}\right)^{\frac{n_1}{2}+k}\left(\frac{n_2}{n_2+n_1x}\right)^{\frac{n_1+n_2}{2}+k}x^{n_1/2-1+k} f X ( x ) = k = 0 ∑ ∞ Beta ( 2 n 2 , 2 n 1 + k ) k ! e − λ /2 ( λ /2 ) k ( n 2 n 1 ) 2 n 1 + k ( n 2 + n 1 x n 2 ) 2 n 1 + n 2 + k x n 1 /2 − 1 + k Función de punto percentil Sample X = ( ∑ i = 1 n 1 ( λ n 1 + Φ − 1 ( u i ) ) 2 ) / n 1 ( 2 P − 1 ( n 2 2 , u ) ) / n 2 \text{Sample}_{X}=\frac{\left(\sum_{i=1}^{n_1}\left(\sqrt{\frac{\lambda}{n_1}}+\Phi^{-1}\left(u_{i}\right)\right)^{2}\right)/n_1}{\left(2\text{P}^{-1}\left(\frac{n_2}{2},u\right)\right)/n_2} Sample X = ( 2 P − 1 ( 2 n 2 , u ) ) / n 2 ( ∑ i = 1 n 1 ( n 1 λ + Φ − 1 ( u i ) ) 2 ) / n 1 Momentos paramétricos no centrados μ k ′ = E [ X k ] = ∫ 0 ∞ x k f X ( x ) d x = e − λ / 2 ( n 1 n 2 ) k Γ ( n 1 / 2 − k ) Γ ( n 1 / 2 ) ∑ r = 0 ∞ ( 1 r ! ) ( λ 2 ) r Γ ( n 1 2 + r + k ) Γ ( n 1 2 + r ) \mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{k}f_{X}\left(x\right)dx=e^{-\lambda/2}\left(\frac{n1}{n2}\right)^{k}\frac{\Gamma\left(n_1/2-k\right)}{\Gamma\left(n_1/2\right)}\sum_{r=0}^{\infty }\left(\frac{1}{r!}\right)\left(\frac{\lambda}{2}\right)^{r}\frac{\Gamma\left(\frac{n_1}{2}+r+k\right)}{\Gamma\left(\frac{n_1}{2}+r\right)} μ k ′ = E [ X k ] = ∫ 0 ∞ x k f X ( x ) d x = e − λ /2 ( n 2 n 1 ) k Γ ( n 1 /2 ) Γ ( n 1 /2 − k ) ∑ r = 0 ∞ ( r ! 1 ) ( 2 λ ) r Γ ( 2 n 1 + r ) Γ ( 2 n 1 + r + k ) Media paramétrica M e a n ( X ) = μ 1 ′ \mathrm{Mean}(X)=\mu'_{1} Mean ( X ) = μ 1 ′ Varianza paramétrica V a r i a n c e ( X ) = μ 2 ′ − μ 1 ′ 2 \mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1} Variance ( X ) = μ 2 ′ − μ 1 ′ 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 ) = 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 : Uniform[0,1] random varible u:\text{Uniform[0,1] random varible} u : Uniform[0,1] random varible 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 P − 1 ( a , u ) : Inverse of regularized lower incomplete gamma function \text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function} P − 1 ( a , u ) : Inverse of regularized lower incomplete gamma function I ( x , a , b ) : Regularized incomplete beta function I\left(x,a,b\right):\text{Regularized incomplete beta function} I ( x , a , b ) : Regularized incomplete beta function Beta ( x , y ) : Beta function \text{Beta}\left(x,y\right):\text{Beta function} Beta ( x , y ) : Beta function