Distribuciones Continuas Distribuciones Continuas
Distribuciones Discretas
ECUACIONES DISTRIBUCIÓN T STUDENT NO CENTRAL Definición de distribución X ∼ N o n C e n t r a l T S t u d e n t ( λ , n , Loc , Sc ) X\sim\mathrm{NonCentralTStudent}\left(\lambda,n,\text{Loc},\text{Sc}\right) X ∼ NonCentralTStudent ( λ , n , Loc , Sc ) Dominio de distribución x ∈ ( − ∞ , ∞ ) x\in\left(-\infty,\infty\right) x ∈ ( − ∞ , ∞ ) Dominio y restricciones de parámetros λ ∈ R , n ∈ R + , Sc ∈ R + , Loc ∈ R \lambda\in\mathbb{R},n\in\mathbb{R}^{+},\text{Sc}\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R} λ ∈ R , n ∈ R + , Sc ∈ R + , Loc ∈ R Función de distribución acumulada F X ( x ) = { 1 2 ∑ j = 0 ∞ 1 j ! ( − λ 2 ) j e − λ 2 2 Γ ( j + 1 2 ) π I n / ( n + z ( x ) 2 ) ( n 2 , j + 1 2 ) if z ( x ) ≥ 0 1 − 1 2 ∑ j = 0 ∞ 1 j ! ( − λ 2 ) j e − λ 2 2 Γ ( j + 1 2 ) π I n / ( n + z ( x ) 2 ) ( n 2 , j + 1 2 ) if z ( x ) < 0 F_{X}\left(x\right)=\left\{\begin{array}{cl}\frac{1}{2}\sum_{j=0}^\infty\frac{1}{j!}(-\lambda\sqrt{2})^je^{\frac{-\lambda^2}{2}}\frac{\Gamma(\frac{j+1}{2})}{\sqrt{\pi}}I_{n/(n+z(x)^2)}\left (\frac{n}{2},\frac{j+1}{2}\right ) & \text{if } \ z(x)\ge 0 \\ 1-\frac{1}{2}\sum_{j=0}^\infty\frac{1}{j!}(-\lambda\sqrt{2})^je^{\frac{-\lambda^2}{2}}\frac{\Gamma(\frac{j+1}{2})}{\sqrt{\pi}}I_{n/(n+z(x)^2)}\left (\frac{n}{2},\frac{j+1}{2}\right ) & \text{if } \ z(x) < 0\end{array} \right. F X ( x ) = ⎩ ⎨ ⎧ 2 1 ∑ j = 0 ∞ j ! 1 ( − λ 2 ) j e 2 − λ 2 π Γ ( 2 j + 1 ) I n / ( n + z ( x ) 2 ) ( 2 n , 2 j + 1 ) 1 − 2 1 ∑ j = 0 ∞ j ! 1 ( − λ 2 ) j e 2 − λ 2 π Γ ( 2 j + 1 ) I n / ( n + z ( x ) 2 ) ( 2 n , 2 j + 1 ) if z ( x ) ≥ 0 if z ( x ) < 0 Función de densidad de probabilidad f X ( x ) = 1 Sc n n / 2 Γ ( n + 1 ) 2 n e λ 2 / 2 ( n + z ( x ) 2 ) n / 2 Γ ( n / 2 ) × { 2 λ z ( x ) 1 F 1 ( n 2 + 1 , 3 2 , λ 2 z ( x ) 2 2 ( n + z ( x ) 2 ) ) ( n + z ( x ) 2 ) Γ ( n + 1 2 ) − 1 F 1 ( n + 1 2 , 1 2 , λ 2 z ( x ) 2 2 ( n + z ( x ) 2 ) ) n + z ( x ) 2 Γ ( n 2 + 1 ) } f_{X}\left(x\right)=\frac{1}{\text{Sc}}\frac{n^{n/2}\Gamma\left(n+1\right)}{2^{n}e^{\lambda^{2}/2}\left(n+z(x)^{2}\right)^{n/2}\Gamma\left(n/2\right)}\times \\ \left\{ \frac{\sqrt{2}\lambda z(x)_{1}F_{1}\left(\frac{n}{2}+1,\frac{3}{2},\frac{\lambda^{2}z(x)^{2}}{2\left(n+z(x)^{2}\right)}\right)}{\left(n+z(x)^{2}\right)\Gamma\left(\frac{n+1}{2}\right)} - \frac{_{1}F_{1}\left(\frac{n+1}{2},\frac{1}{2},\frac{\lambda^{2}z(x)^{2}}{2\left(n+z(x)^{2}\right)}\right)}{\sqrt{n+z(x)^{2}}\Gamma\left(\frac{n}{2}+1\right)} \right\} f X ( x ) = Sc 1 2 n e λ 2 /2 ( n + z ( x ) 2 ) n /2 Γ ( n /2 ) n n /2 Γ ( n + 1 ) × ⎩ ⎨ ⎧ ( n + z ( x ) 2 ) Γ ( 2 n + 1 ) 2 λ z ( x ) 1 F 1 ( 2 n + 1 , 2 3 , 2 ( n + z ( x ) 2 ) λ 2 z ( x ) 2 ) − n + z ( x ) 2 Γ ( 2 n + 1 ) 1 F 1 ( 2 n + 1 , 2 1 , 2 ( n + z ( x ) 2 ) λ 2 z ( x ) 2 ) ⎭ ⎬ ⎫ Función de punto percentil Sample X = Loc + Sc ( λ + Φ − 1 ( u ) ) ( 2 P − 1 ( n 2 , u ) ) / n \text{Sample}_{X}=\text{Loc}+\text{Sc}\frac{\left(\lambda+\Phi^{-1}\left(u\right)\right)}{\left(\sqrt{2\text{P}^{-1}\left(\frac{n}{2},u\right)}\right)/n} Sample X = Loc + Sc ( 2 P − 1 ( 2 n , u ) ) / n ( λ + Φ − 1 ( u ) ) Momentos paramétricos no centrados μ ~ k ′ = E [ X ~ k ] = ∫ 0 ∞ x k f X ~ ( x ) d x = e − λ 2 / 2 n π Γ ( n / 2 ) Γ ( n − k 2 ) n k / 2 ∑ r = 0 ∞ λ r 2 r / 2 r ! Γ ( r + k + 1 2 ) \tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{e^{-\lambda^{2}/2}}{\sqrt{n\pi}\Gamma\left(n/2\right)}\Gamma\left(\frac{n-k}{2}\right)n^{k/2}\sum_{r=0}^{\infty }\frac{\lambda^{r}2^{r/2}}{r!}\Gamma\left(\frac{r+k+1}{2}\right) μ ~ k ′ = E [ X ~ k ] = ∫ 0 ∞ x k f X ~ ( x ) d x = nπ Γ ( n /2 ) e − λ 2 /2 Γ ( 2 n − k ) n k /2 ∑ r = 0 ∞ r ! λ r 2 r /2 Γ ( 2 r + k + 1 ) Media paramétrica M e a n ( X ) = Loc + Sc μ ~ 1 ′ \mathrm{Mean}(X)=\text{Loc}+\text{Sc}\tilde{\mu}'_{1} Mean ( X ) = Loc + Sc μ ~ 1 ′ Varianza paramétrica V a r i a n c e ( X ) = Sc 2 ( μ ~ 2 ′ − μ ~ 1 ′ 2 ) \mathrm{Variance}(X)=\text{Sc}^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}) Variance ( X ) = Sc 2 ( 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{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\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{\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}} 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. X ~ ∼ N o n C e n t r a l T S t u d e n t ( λ , n , 0 , 1 ) \tilde{X}\sim\mathrm{NonCentralTStudent}\left(\lambda,n,0,1\right) X ~ ∼ NonCentralTStudent ( λ , n , 0 , 1 ) Loc : Location parameter \text{Loc}:\text{Location parameter} Loc : Location parameter Sc : Scale parameter \text{Sc}:\text{Scale parameter} Sc : Scale parameter z ( x ) = ( x − Loc ) / Sc z\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc} z ( x ) = ( x − Loc ) / Sc u : Uniform[0,1] random varible u:\text{Uniform[0,1] random varible} u : Uniform[0,1] random varible 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 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 1 F 1 ( a , b , z ) : Kummer’s confluent hypergeometric function _{1}F_{1}(a,b,z):\text{Kummer's confluent hypergeometric function} 1 F 1 ( a , b , z ) : Kummer’s confluent hypergeometric function