Probability

Covariance

Whether deviations from the mean tend to share a sign

Joint distribution ($\sigma_x = \sigma_y = 1$, zero-mean)

Σ

1 1

Key ideas

  • $\text{cov}(X,Y) = \mathbb{E}[(X-\mathbb{E}[X])\cdot(Y-\mathbb{E}[Y])]$. Positive when deviations share a sign, negative when they oppose.
  • $\rho = \frac{\text{cov}(X,Y)}{\sigma_x\sigma_y}$ is the dimensionless version, always between −1 and 1.
  • $\text{cov}(X,Y) = 0$ does not imply independence. X and Y can be dependent with zero covariance (e.g., $Y = X^2$).
  • Independence does imply zero covariance, but not the other way around.