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].