Statistik 3 - Beispiel 1

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Angabe: media:StatIII ex 1-5.pdf

Bsp. 1

A

 
 s_{x,y} =
 \frac 1 n \sum_{i=1}^n (x_i - \bar{x}) y_i = 
 \frac 1 n \sum_{i=1}^n x_i \cdot y_i - \bar{x} \cdot y_i = 
 \frac 1 n \left( \sum_{i=1}^n x_i \cdot y_i - \sum_{i=1}^n \bar{x} \cdot y_i \right) =


 \frac 1 n \left( \sum_{i=1}^n x_i \cdot y_i - \bar{x} \sum_{i=1}^n y_i \right) =

 
 = \frac 1 n \left( \sum_{i=1}^n x_i \cdot y_i \right) - \bar{x} \frac 1 n \sum_{i=1}^n y_i =
 \frac 1 n \left( \sum_{i=1}^n x_i \cdot y_i \right) - \bar{x} \bar y

B


 s_{\alpha x, \beta y} = 
 \frac{\frac 1 n \sum_{i=1}^n (\alpha x_i - \bar{\alpha x_i}) (\beta y_i - \bar{\beta y_i}) }
      {\sqrt{ s_{\alpha x} \cdot s_{\beta y} }} =


 \frac{\frac 1 n \sum_{i=1}^n \alpha \beta (x_i - \bar{x_i}) (y_i - \bar{y_i}) }
      {\sqrt{ \alpha^2 s_{x} \cdot \beta^2 s_{y} }} =


 \frac{\alpha \beta \frac 1 n \sum_{i=1}^n (x_i - \bar{x_i}) (y_i - \bar{y_i}) }
      {\alpha \beta \sqrt{ s_{x} \cdot s_{y} }} =


 \frac{\frac 1 n \sum_{i=1}^n (x_i - \bar{x_i}) (y_i - \bar{y_i}) }
      {\sqrt{ s_{x} \cdot s_{y} }}

C

Wieso ist, falls s_{x,x} > 0 \, und s_{y,y} > 0 , -1 \le r_{x,y} \le 1?

Mit Hilfe der Cauchy-Schwarzen Ungleichung

-1 \le r_{x,y} \le 1

-1 \le \frac{s_{x,y}}{\sqrt{s_x s_y}} \le 1

| \frac{s_{x,y}}{\sqrt{s_x s_y}} | \le 1

| s_{x,y} | \le \sqrt{s_x s_y}

 s_{x,y}^2 \le | s_x s_y |

 s_{x,y}^2 \le | s_x | | s_y |

 \langle x,y \rangle^2 \le |\langle x, x \rangle | |\langle y, y \rangle |

 | \langle x,y \rangle | \le |x| |y|