第1講確率と確率変数

相関係数

ピアソンの積率相関係数は，

\begin{aligned} r_{X Y} & = \frac{C o v [X, Y]}{\sqrt{V [X]} \sqrt{V [Y]}} \\ = \frac{\sum_{i = 1}^{n} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{\sqrt{\sum_{i = 1}^{n} (x_{i} - \bar{x})^{2}} \sqrt{\sum_{i = 1}^{n} (y_{i} - \bar{y})^{2}}} \end{aligned}

$\begin{align} r_{XY} &= \frac{Cov[X, Y]}{\sqrt{V[X]}\sqrt{V[Y]}} \\ &= \frac{ \displaystyle \sum_{i=1}^{n} (x_{i}-\bar{x})(y_{i}-\bar{y}) }{ \displaystyle \sqrt{\sum_{i=1}^n(x_{i}-\bar{x})^2} \sqrt{\sum_{i=1}^n(y_{i}-\bar{y})^2}} \end{align}$

単に相関係数といえば，ピアソンの積率相関係数を指す．

ここで， $-1 \le r_{XY} \le 1$ となることを示す．シュワルツの不等式より，次の式が成り立つ．

{\sum_{i = 1}^{n} (x_{i} - \bar{x}) (y_{i} - \bar{y})}^{2} \leq (\sum_{i = 1}^{n} (x_{i} - \bar{x})^{2}) (\sum_{i = 1}^{n} (y_{i} - \bar{y})^{2})

$\begin{equation} \left\{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})\right\}^2 \le \left(\sum_{i=1}^{n}(x_i - \bar{x})^2\right)\left(\sum_{i=1}^{n}(y_i - \bar{y})^2\right) \end{equation}$

これより，

{r_{X Y}}^{2} = \frac{{\sum_{i = 1}^{n} (x_{i} - \bar{x}) (y_{i} - \bar{y})}^{2}}{(\sum_{i = 1}^{n} (x_{i} - \bar{x})^{2}) (\sum_{i = 1}^{n} (y_{i} - \bar{y})^{2})} \leq 1

$\begin{equation} {r_{XY} }^2 = \frac{\ds \left\{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})\right\}^2}{\ds \left(\sum_{i=1}^{n}(x_i - \bar{x})^2\right)\left(\sum_{i=1}^{n}(y_i - \bar{y})^2\right)} \le 1 \end{equation}$

よって，

- 1 \leq r_{X Y} \leq 1

$\begin{equation} -1 \le r_{XY} \le 1 \end{equation}$

第1講 確率と確率変数

相関係数

第1講確率と確率変数