(統計)共變異數(Covariance)，相關係數(correlation coefficient)

共變異數(Covariance)

$$cov(X,Y)=E[(X-\mu_x)(X-\mu_y)]=E(XY)-\mu_x \mu_y$$

$cov(X,X)=V(X)$

$cov(X,Y)=cov(Y,X)$

$cov(aX+b,cY+d)=ac\ cov(X,Y)$

$cov(X,Y+Z)=cov(X,Y)+cov(X,Z)$

若$T=\sum_{i=1}^na_iX_i\ \ ,for\ i=1,\cdots,n$

$$var(T)=\sum_{i=1}^na_i^2var(X_i)+2\sum_{i<j}a_ia_jcov(X_i,X_j)$$

若$T=aX+bY+cZ$

$var(T)=a^2var(X)+b^2var(Y)+c^2var(Y)+2abcov(X,Y)+2bccov(Y,Z)+2accov(X,Z)$

$var(aX\pm bY)=a^2var(X)+b^2var(Y)\pm 2abcov(X,Y)$

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相關係數(correlation coefficient)

$$\rho(X,Y)=\cfrac{cov(X,Y)}{\sigma_1\sigma_2}$$

若 $X,Y$ 獨立，則 $cov(X,Y)=0=\rho$

$若\rho=0$，則 $X，Y$未必獨立。

$\rho(X+a,Y+b)=\rho(X,Y)$

$\rho(X,aY)=\left\{ \begin{array}{c} \rho(X,Y)   &, if \ a\gt0\\ -\rho(X,Y)  &, if \ a\lt0  \end{array}\right.$

$\rho(aX+b,cY+d)=\left\{ \begin{array}{c} \rho(X,Y)   &, if \ \ ac\gt0\\ -\rho(X,Y)  &, if \ \ ac\lt0  \end{array}\right.$

$-1\leq \rho(X,Y)\leq 1$  $\left\{ \begin{array}{l} \rho(X,Y)=1   &, (完全正相關)\\ \rho(X,Y)=-1  &, (完全負相關)  \\\rho(X,Y)=0 &, (零相關)\end{array}\right.$

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證 $-1\leq \rho \leq1$

考慮 $var(\cfrac{X}{\sigma_X}-\cfrac{Y}{\sigma_Y})\geq 0\implies \cfrac{1}{\sigma_X^2}var(X)+\cfrac{1}{\sigma_Y^2}var(Y)-2\cfrac{1}{\sigma_X}\cfrac{1}{\sigma_Y}cov(X,Y)\geq 0$

$\implies 1+1-2\rho_{X,Y}\geq 0\implies\rho_{X,Y}\leq 1\ \cdots\ (1)$

$var(\cfrac{X}{\sigma_X}+\cfrac{Y}{\sigma_Y})\geq 0\implies\rho_{X,Y}\geq -1\ \cdots\ (2)$

由 $(1) ,(2) 得 -1\leq \rho \leq1$