Fórmulas en Latex


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A

ANOVA fuentes de varianza

{\displaystyle \sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\left(y_{ij}-\bar{y}\right)^{2}=n\sum_{i=1}^{k}\left(\bar{y}_{i}-\bar{y}\right)^{2}+\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\left(y_{ij}-\bar{y}_{i}\right)^{2}}

$${\displaystyle \sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\left(y_{ij}-\bar{y}\right)^{2}=n\sum_{i=1}^{k}\left(\bar{y}_{i}-\bar{y}\right)^{2}+\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\left(y_{ij}-\bar{y}_{i}\right)^{2}}$$


ANOVA modelo una via

\large y_{ij}=\mu+\tau_{i}+\varepsilon_{ij}
$$\large y_{ij}=\mu+\tau_{i}+\varepsilon_{ij}$$

C

Componentes principales (cargas)


Componentes principales (ejes)


D

Distancia de Mahalanobis

d_{m}\left(\underset{\sim}{x}_{i},\underset{\sim}{x}_{j}\right)=\sqrt{\left(\underset{\sim}{x}_{i}-\underset{\sim}{x}_{j}\right)^{t}\sum^{-1}\left(\underset{\sim}{x}_{i}-\underset{\sim}{x}_{j}\right)}

$$d_{m}\left(\underset{\sim}{x}_{i},\underset{\sim}{x}_{j}\right)=\sqrt{\left(\underset{\sim}{x}_{i}-\underset{\sim}{x}_{j}\right)^{t}\sum^{-1}\left(\underset{\sim}{x}_{i}-\underset{\sim}{x}_{j}\right)}$$


E

ecuación varias lineas

\begin{eqnarray}
 y &=& x^4 + 4      \nonumber \\
   &=& (x^2+2)^2 -4x^2 \nonumber \\
   &\le&(x^2+2)^2
\end{eqnarray}

$$\begin{eqnarray}
 y &=& x^4 + 4      \nonumber \\
   &=& (x^2+2)^2 -4x^2 \nonumber \\
   &\le&(x^2+2)^2
\end{eqnarray}$$

F

fórmula en linea

x=\frac{1+y}{1+2z^2}

(en linea)

$$x=\frac{1+y}{1+2z^2}$$


fórmula varios niveles


 \frac{1}{\displaystyle 1+
   \frac{1}{\displaystyle 2+
   \frac{1}{\displaystyle 3+x}}} +
 \frac{1}{1+\frac{1}{2+\frac{1}{3+x}}}
$$
 \frac{1}{\displaystyle 1+
   \frac{1}{\displaystyle 2+
   \frac{1}{\displaystyle 3+x}}} +
 \frac{1}{1+\frac{1}{2+\frac{1}{3+x}}}
$$

R

Regresión Múltiple (matrices)

\overset{\wedge}{\underset{\sim}{\beta}}=\left(\begin{array}{ccccc}\hat{\beta_{0}}\\\hat{\beta_{1}}\\\hat{\beta_{2}}\\\vdots\\\hat{\beta_{p}}\end{array}\right)=\left(\begin{array}{ccccc}x_{11} & x_{21} & x_{31} & \cdots & x_{n1}\\x_{21} & x_{22} & x_{32} & \cdots & x_{n2}\\x_{13} & x_{23} & x_{33} & \cdots & x_{n3}\\\vdots & \vdots & \vdots & \ddots & \vdots\\x_{1p} & x_{2p} & x_{3p} & \cdots & x_{np}\end{array}\right)\left(\begin{array}{ccccc}x_{11} & x_{12} & x_{13} & \cdots & x_{1p}\\x_{21} & x_{22} & x_{23} & \cdots & x_{2p}\\x_{31} & x_{32} & x_{33} & \cdots & x_{3p}\\\vdots & \vdots & \vdots & \ddots & \vdots\\x_{n1} & x_{n2} & x_{n3} & \cdots & x_{np}\end{array}\right)\left(\begin{array}{ccccc}y_{1}\\y_{2}\\y_{3}\\\vdots\\y_{n}\end{array}\right)
$$\overset{\wedge}{\underset{\sim}{\beta}}=\left(\begin{array}{ccccc}
\hat{\beta_{0}}\\
\hat{\beta_{1}}\\
\hat{\beta_{2}}\\
\vdots\\
\hat{\beta_{p}}\end{array}\right)=
\left(\begin{array}{ccccc}
x_{11} & x_{21} & x_{31} & \cdots & x_{n1}\\
x_{21} & x_{22} & x_{32} & \cdots & x_{n2}\\
x_{13} & x_{23} & x_{33} & \cdots & x_{n3}\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
x_{1p} & x_{2p} & x_{3p} & \cdots & x_{np}\end{array}\right)
\left(\begin{array}{ccccc}
x_{11} & x_{12} & x_{13} & \cdots & x_{1p}\\
x_{21} & x_{22} & x_{23} & \cdots & x_{2p}\\
x_{31} & x_{32} & x_{33} & \cdots & x_{3p}\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
x_{n1} & x_{n2} & x_{n3} & \cdots & x_{np}\end{array}\right)
\left(\begin{array}{ccccc}
y_{1}\\
y_{2}\\
y_{3}\\
\vdots\\
y_{n}\end{array}\right)$$
 

V

Varianza muestral


s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}

 

$$s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}$$