Covariance

Study guides: Exam P-1, Exam FAM, Exam MAS-I

Covariance measures the linear association between two random variables $X$ and $Y$ . Positive covariance indicates the variables tend to move together; negative covariance indicates they tend to move in opposite directions.

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

$= E[(X - \mu_X)(Y - \mu_Y)]$

If $X$ and $Y$ are independent, $\text{Cov}(X, Y) = 0$ (but the converse is not necessarily true)

Computing Covariance from Expectations Given $E[X] = 2$ , $E[Y] = 4$ , and $E[XY] = 10$ , what is $\text{Cov}(X, Y)$ ?

Answer $\text{Cov}(X, Y) = E[XY] - E[X] \cdot E[Y] = 10 - 2 \times 4 = 10 - 8 = 2$ The positive covariance suggests $X$ and $Y$ tend to increase together.