Covariance and Correlation: Measuring Dependence for Exam P

Covariance

The covariance of X and Y is Cov(X, Y) = E[XY] - E[X]*E[Y]. If X and Y are independent, Cov(X, Y) = 0 (but the converse is false). Positive covariance means X and Y tend to move together; negative covariance means they tend to move in opposite directions.

Key properties: Cov(X, X) = Var(X), Cov(aX + b, cY + d) = ac * Cov(X, Y), and Var(X + Y) = Var(X) + Var(Y) + 2*Cov(X, Y). For n variables, Var(sum Xi) = sum Var(Xi) + 2 * sum_{i

Correlation

The correlation coefficient rho(X, Y) = Cov(X, Y) / (sigma_X * sigma_Y) standardizes covariance to lie in [-1, 1]. rho = 1 or -1 indicates a perfect linear relationship: Y = aX + b with a > 0 (rho = 1) or a < 0 (rho = -1). rho = 0 means uncorrelated, which is weaker than independence.

On Exam P, you might be given a joint distribution and asked to compute the correlation. The steps are: find E[X], E[Y], E[XY], E[X^2], E[Y^2], then compute Cov, Var(X), Var(Y), and finally rho.

Exam P Pitfalls

A common Exam P trap involves the statement "zero covariance implies independence." This is true only for jointly normal random variables, not in general. Also remember that for indicator random variables I_A and I_B, Cov(I_A, I_B) = P(A and B) - P(A)*P(B). This makes covariance computations straightforward for event-based problems. Practice recognizing when the shortcut formula E[XY] - E[X]*E[Y] is more efficient than the definition E[(X - mu_X)(Y - mu_Y)].