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How to compute correlation from variances

JefferiesData Scientist · Technical Screen
EasyData ScientistTechnical ScreenStatistics & Math

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Practice deriving correlation from variance identities for two random variables. The solution explains why Var(X+Y) and Var(X) alone are insufficient, derives covariance and correlation when Var(Y) is known, simplifies the equal-variance case, and checks edge cases where correlation is invalid or inconsistent.

Statistics Prompt: Compute Correlation from Variances

You have two random variables (X) and (Y). You are told:

  • (\mathrm{Var}(X+Y))
  • (\mathrm{Var}(X))

Answer the following:

  1. Is this information sufficient to determine the correlation (\rho_{X,Y})? If not, what additional quantity or quantities would you need?
  2. If you also know (\mathrm{Var}(Y)), derive (\rho_{X,Y}) in terms of (\mathrm{Var}(X+Y)), (\mathrm{Var}(X)), and (\mathrm{Var}(Y)).
  3. Special case: if you assume (\mathrm{Var}(Y)=\mathrm{Var}(X)), simplify the expression for (\rho_{X,Y}).

Constraints & Assumptions

  • Assume (X) and (Y) have finite, positive variances unless you explicitly discuss a degenerate case.
  • Use the variance identity for sums and the definition of correlation.
  • Call out when a proposed set of variance values is internally inconsistent.

Clarifying Questions to Ask

  • Are (\mathrm{Var}(X)) and (\mathrm{Var}(Y)) both positive?
  • Are the given variances exact population values or sample estimates?
  • Is the question asking for a symbolic derivation or a numeric example?

Follow-up questions

  • What happens if \(\mathrm{Var}(X)=0\) or \(\mathrm{Var}(Y)=0\)?
  • How would the formula change for \(\mathrm{Var}(X-Y)\)?
  • Can you provide two examples with the same \(\mathrm{Var}(X+Y)\) and \(\mathrm{Var}(X)\) but different correlations?
  • How would sample estimation error affect the computed correlation?