Covariance and Correlation

Probability · Axiom Academy

LESSON Covariance and Correlation Measuring the linear relationship between random variables 1. Covariance: Joint Variability The covariance between random variables X and Y measures how much they vary together from their means. • Cov(X, Y) > 0: X and Y tend to increase together (positive relationship) • Cov(X, Y) < 0: when X increases, Y tends to decrease (negative relationship) • Cov(X, Y) = 0: no linear relationship (but may still be dependent!) 2. Correlation: Standardized Covariance The correlation coefficient ρ (rho) is covariance normalized by the standard deviations, giving a dimensionless measure of linear association. Correlation always satisfies -1 ≤ ρ ≤ +1: • ρ = +1: perfect positive linear relationship • ρ = -1: perfect negative linear relationship • ρ = 0: no linear relationship 3. Independence vs Zero Correlation If X and Y are independent , then Cov(X, Y) = 0 and ρ = 0. However, the converse is NOT true: ρ = 0 does NOT imply independence! Example: Let X ~ Uniform -1, 0, 1 and Y = X². Then ρ = 0 (no linear relationship), but X and Y are clearly dependent (Y is completely determined by X). Zero correlation only means no linear relationship. Variables can still have strong nonlinear relationships.

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