Covariance and Correlation

Compact study note.

Summary

Covariance measures linear co-movement between two random variables. Correlation rescales covariance to a unitless number between $-1$ and $1$ when both variances are positive.^[1]

Prerequisites

• Expected Value
• Variance

Notation and Assumptions

Core definitions:

Essential Result

$\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)+2\mathrm{Cov}(X,Y)$ . Independence implies zero covariance when moments exist, but zero covariance does not imply independence in general.

Small Example

If $Y=2X+1$ and $\mathrm{Var}(X)>0$ , then ${\rho }_{X,Y}=1$ because the relationship is exactly increasing linear.

Common Mistakes

• Equating zero correlation with independence.
• Forgetting units: covariance has product units; correlation has no units.

Connections

• Independence
• Variance
• Joint Marginal and Conditional Distributions

References