# Deriving Correlation from Variances You have two random variables X and Y with finite second moments. You are given Var(X + Y) and Var(X). Answer the questions below. ### Constraints & Assumptions - All variances are computed under the same probability model. - Variances are finite. - Correlation is defined only when Var(X) and Var(Y) are positive. - Show the variance identity used in the derivation. ### Clarifying Questions to Ask - Is Var(Y) known? - Are X and Y non-degenerate? - Are we assuming independence, or should covariance be inferred? - Are these population variances or sample estimates? ### Part 1 - Identifiability Is Corr(X, Y) identifiable from only Var(X + Y) and Var(X)? Explain why or why not. #### What This Part Should Cover - Use Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X, Y). - Explain that Var(Y) and Cov(X, Y) are both unknown. - State that one equation with two unknowns does not identify correlation. - Note that covariance must be normalized by both standard deviations. ### Part 2 - Derivation with Var(Y) If Var(Y) is also given, derive Corr(X, Y) in terms of Var(X + Y), Var(X), and Var(Y). #### What This Part Should Cover - Rearrange the variance identity to get Cov(X, Y) = [Var(X + Y) - Var(X) - Var(Y)] / 2. - Divide by sqrt(Var(X) Var(Y)). - Provide the final formula for correlation. - Mention valid inputs must produce a correlation in [-1, 1]. ### Follow-up Questions - How would the formula change for Var(X - Y)? - What if Var(Y) = 0? - How would sampling error affect an estimated correlation from sample variances?