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Questions about covariance, a measure of (linear) association between two random variables.

Covariance is a measure which shows how much two RVs are dependent. If they are fully independent it would be zero and as much as they are dependent it would have a greater value. You can have a much powerful insight by description of the following formula:

The covariance of the random variables $X$ and $Y$ is the difference of the Expected value of their product ($E(XY)$) by the product of their expected values ($E(X)E(Y)$).

\begin{align*} \sigma(X,Y) = E(XY)-E(X)E(Y) \end{align*}

If they are independent then $E(XY)=E(X)E(Y)$ and therefore the covariance would be zero. Also, as much as they depend on each other their distance would be higher.

Though the main formula for definition of co-variance is \begin{align*} \sigma(X,Y) = E \left[ \left(X-E(X)\right) \left(Y-E(Y)\right) \right] \end{align*}

we can convert it to the pre-explained one (for the finite-domain random variables):

\begin{align*} \sigma(X,Y) &= E \left[ \left(X-E(X)\right) \left(Y-E(Y)\right) \right] \\\ &= E \left[ X Y - X E(Y) - E(X) Y + E(X) E(Y) \right]\\\ &= E (X Y) - E(X) E(Y) - E(X) E(Y) + E(X) E(Y) \\\ &= E (X Y) - E(X) E(Y) \end{align*}

Also, for two vectors of random variables ($\mathbb{X}$ and $\mathbb{Y}$) the covariance matrix has been defined as a matrix which each cell shows the covariance of corresponding cell in the matrix ($\mathbb{X} \times \mathbb{Y}^T$).

Reference:

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