Covariance, Variance of Sums, and Correlations

Definition Let $\mathbb{X}, \mathbb{Y}$ be two random variables. The covariance of $\mathbb{X}, \mathbb{Y}$ is defined as $\mathbb{E}[(\mathbb{X} - \mathbb{E}[\mathbb{X}])(\mathbb{Y} - \mathbb{E}[\mathbb{Y}])]$

Simple rearrangement shows $cov( \mathbb{X}, \mathbb{Y} ) = \mathbb{E}[\mathbb{X}\mathbb{Y}] - \mathbb{E}[\mathbb{X}]\mathbb{E}[\mathbb{Y}]$

Definition the correlation of $\mathbb{X}, \mathbb{Y}$, $\rho_{XY}$ is $\frac{cov( \mathbb{X}, \mathbb{Y} )}{\sqrt{var(X)}\sqrt{var(Y)}}$

Property or covariance

1. if $\mathbb{X}$ and $\mathbb{Y}$ are independent, $cov( \mathbb{X}, \mathbb{Y} ) = 0$
2. $var(\sum_{i=1}^n X_i) = \mathbb{E}[(\sum_{i=1}^n X_i - \mathbb{E}[\sum_{i=1}^n X_i])^2]$ $= \sum_{i=1}^n \mathbb{E}[ (X_i - \mathbb{E}[X_i])^2 ] - \sum_{i!=j}\mathbb{E}[ (\mathbb{X}_i - \mathbb{E}[\mathbb{X}_i])(\mathbb{X}_j - \mathbb{E}[\mathbb{X}_j]) ]$ $= \sum_{i=1}^n var(X_i) + 2 \sum_{i<j}cov(X_i, X_j)$
3. By (b), if $X_1, ..., X_n$ are independent, $var(\sum_{i=1}^n X_i) = \sum_{i=1}^n var(X_i)$