Variance and Covariance Properties

$$\sigma_X^2 := \text{Var}(X) = E\left[(X - E[X])^2\right] = E(X^2) - (E(X))^2$$

$$\sigma_X := \text{SD}(X) = \sqrt{\text{Var}(X)}$$

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

$$\rho(X, Y) = \text{Corr}(X, Y) = \frac{\text{Cov}(X, Y)}{\text{SD}(X)\,\text{SD}(Y)}$$

$\text{Var}(aX + b) = a^2 \text{Var}(X)$
$\text{SD}(aX + b) = |a|\,\text{SD}(X)$
$\text{Cov}(X, Y) = \text{Cov}(Y, X)$
$\text{Cov}(aX + b,\; cY + d) = ac\,\text{Cov}(X, Y)$
$\text{Cov}(X, X) = \text{Var}(X)$
$\text{Cov}\!\left(\sum_{i=1}^n a_i X_i,\;\sum_{j=1}^m b_j Y_j\right) = \sum_{i=1}^n \sum_{j=1}^m a_i b_j \,\text{Cov}(X_i, Y_j)$
$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\,\text{Cov}(X, Y)$
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y) - 2\,\text{Cov}(X, Y)$

$$\text{Var}\!\left(\sum_{i=1}^n a_i X_i\right) = \sum_{i=1}^n a_i^2 \text{Var}(X_i) + 2\sum_{i < j} a_i a_j \,\text{Cov}(X_i, X_j)$$

If $X$ and $Y$ are independent, then $\text{Cov}(X, Y) = 0$, so $\text{Var}(X \pm Y) = \text{Var}(X) + \text{Var}(Y)$.

Variance and Covariance Properties#