covariance & correlation

14 Feb 2018 on Math

1. sample covariance

$s_{xy} = \dfrac{1}{N}\sum_{i=1}^{N} (x_i-m_x)(y_i-m_y)$

2. sample correlation coefficient

$r_{xy} = \dfrac{s_{xy}}{\sqrt{s^2_{x} \cdot s^2_{y}}}$

3. covariance

$\text{Cov}[X, Y] = \text{E}[(X - \text{E}[X])(Y - \text{E}[Y])]$

4. correlation coefficient

$\rho[X,Y] = \dfrac{\text{Cov}[X, Y]}{\sqrt{\text{Var}[X] \cdot \text{Var}[Y]}}$

$-1 \leq \rho \leq 1$

Reference