PLAYGROUND

ECUACIONES DISTRIBUCIÓN T STUDENT

Definición de distribución

X∼TStudent(df)X\sim\mathrm{TStudent}\left(\text{df}\right)\\

Dominio de distribución

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Dominio y restricciones de parámetros

df∈R+\text{df}\in\mathbb{R}^{+}\\

Función de distribución acumulada

FX(x)=I(x+x2+df2x2+df,df2,df2)F_{X}\left(x\right)=I\left(\frac{x+\sqrt{x^{2}+\text{df}}}{2\sqrt{x^{2}+\text{df}}},\frac{\text{df}}{2},\frac{\text{df}}{2}\right)\\

Función de densidad de probabilidad

fX(x)=(1+x2/df)−(1+df)/2df×Beta(12,df2)f_{X}\left(x\right)=\frac{\left(1+x^{2}/\text{df}\right)^{-(1+\text{df})/2}}{\sqrt{\text{df}}\times \text{Beta}\left(\frac{1}{2},\frac{\text{df}}{2}\right)}\\

Función de punto percentil

FX−1(u)={df(1−I−1(u,df/2,df/2))I−1(u,df/2,df/2)if  u≥12−df(1−I−1(u,df/2,df/2))I−1(u,df/2,df/2)if  u<12F^{-1}_{X}\left(u\right)=\left\{\begin{array}{cl} \sqrt{\frac{\text{df}(1-I^{-1}\left(u,\text{df}/2,\text{df}/2\right))}{I^{-1}\left(u,\text{df}/2,\text{df}/2\right)}} & \text{if } \ u \geq \frac{1}{2} \\ -\sqrt{\frac{\text{df}(1-I^{-1}\left(u,\text{df}/2,\text{df}/2\right))}{I^{-1}\left(u,\text{df}/2,\text{df}/2\right)}} & \text{if } \ u < \frac{1}{2} \end{array} \right.\\

Momentos paramétricos no centrados

μk′=E[Xk]=∫−∞∞xkfX(x)dx={0if  k odd ∧ 0<k<dfdfk2 ∏i=1k/22i−1df−2iif  k even ∧ 0<k<df\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx=\left\{\begin{array}{cl} 0 & \text{if } \ k\text{ odd} \ \wedge \ 0 < k < \text{df} \\ \text{df}^{\frac{k}{2}} \,\prod_{i=1}^{k/2}\frac{2i-1}{\text{df}-2i} & \text{if } \ k\text{ even} \ \wedge \ 0 < k < \text{df} \end{array} \right.\\

Media paramétrica

Mean(X)=μ1′=0\mathrm{Mean}(X)=\mu'_{1}=0\\

Varianza paramétrica

Variance(X)=μ2′−μ1′2={df/(df+2)if  df>2undefinedif  df≤2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\left\{\begin{array}{cl} \text{df}/(\text{df}+2) & \text{if } \ \text{df} > 2 \\ \text{undefined} & \text{if } \ \text{df} \leq 2 \end{array} \right.\\

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5={0if  df>3undefinedif  df≤3\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\left\{\begin{array}{cl} 0 & \text{if } \ \text{df} > 3 \\ \text{undefined} & \text{if } \ \text{df} \leq 3 \end{array} \right.\\

Curtosis paramétrica

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2={3+6/(df−4)if  df>4undefinedif  df≤4\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}}=\left\{\begin{array}{cl} 3+6/(\text{df}-4) & \text{if } \ \text{df} > 4 \\ \text{undefined} & \text{if } \ \text{df} \leq 4 \end{array} \right.\\

Mediana paramétrica

Median(X)=0\mathrm{Median}(X)=0\\

Moda paramétrica

Mode(X)=0\mathrm{Mode}(X)=0

Información y definiciones adicionales

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
I−1(x,a,b):Inverse of regularized incomplete beta functionI^{-1}\left(x,a,b\right):\text{Inverse of regularized incomplete beta function}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}