ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Binary cartesian set product ▼ Binary relation ▼ Map ▼ Simple map ▼ Simple function ▼ Measurable simple complex function ▼ Simple integral ▼ Unsigned basic integral ▼ Unsigned basic expectation ▼ Basic expectation ▼ Random real number moment ▼ Random real number central moment ▼ Joint central moment
Definition D2011
Euclidean real covariance

Let $X, Y \in \text{Random}(\mathbb{R}^{N \times 1})$ each be a D5210: Random real column matrix such that
 (i) $$\mathbb{E} |X|^2, \mathbb{E} |Y|^2 < \infty$$
The covariance of $(X, Y)$ is the D4571: Real matrix $$\mathbb{E} \left[ (X - \mathbb{E} X) (Y - \mathbb{E} Y)^T \right]$$

Let $X, Y \in \text{Random}(\mathbb{R}^N)$ each be a D4383: Random euclidean real number such that
 (i) $$\mathbb{E} |X|^2, \mathbb{E} |Y|^2 < \infty$$
The covariance of $(X, Y)$ is the D4571: Real matrix $$\begin{bmatrix} \mathbb{E}[(X_1 - \mathbb{E} X_1) (Y_1 - \mathbb{E} Y_1)] & \mathbb{E}[(X_1 - \mathbb{E} X_1) (Y_2 - \mathbb{E} Y_2)] & \cdots & \mathbb{E}[(X_1 - \mathbb{E} X_1) (Y_N - \mathbb{E} Y_N)] \\ \mathbb{E}[(X_2 - \mathbb{E} X_2) (Y_1 - \mathbb{E} Y_1)] & \mathbb{E}[(X_2 - \mathbb{E} X_2) (Y_2 - \mathbb{E} Y_2)] & \vdots & \mathbb{E}[(X_2 - \mathbb{E} X_2) (Y_N - \mathbb{E} Y_N)] \\ \vdots & \cdots & \ddots & \vdots \\ \mathbb{E}[(X_N - \mathbb{E} X_N) (Y_1 - \mathbb{E} Y_1)] & \mathbb{E}[(X_N - \mathbb{E} X_N) (Y_2 - \mathbb{E} Y_2)] & \cdots & \mathbb{E}[(X_N - \mathbb{E} X_N) (Y_N - \mathbb{E} Y_N)] \end{bmatrix}$$
Children
 ▶ Euclidean real variance