How to derive correlation from variances
Problem Statement
Consider two random variables, X and Y.
You are initially provided with the following information:
- The variance of their sum: Var(X + Y)
- The variance of X: Var(X)
Questions
- Identifiability: Can the correlation coefficient, Corr(X, Y), be determined solely from Var(X + Y) and Var(X)? Provide a clear explanation for your answer.
- Derivation: If, in addition to the above, you are also given Var(Y), derive the expression for Corr(X, Y) using Var(X + Y), Var(X), and Var(Y).
Assumptions
Assume that both random variables possess finite second moments, and all variances are computed under the same underlying probability model.
AI interview prep
Core dilemma
Identifying the minimum set of statistical moments required to uniquely determine the correlation between two random variables, and demonstrating the algebraic derivation with rigor.
2 tradeoffs and 3 stress tests locked
Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
How to derive correlation from variances
Problem Statement
Consider two random variables, X and Y.
You are initially provided with the following information:
- The variance of their sum: Var(X + Y)
- The variance of X: Var(X)
Questions
- Identifiability: Can the correlation coefficient, Corr(X, Y), be determined solely from Var(X + Y) and Var(X)? Provide a clear explanation for your answer.
- Derivation: If, in addition to the above, you are also given Var(Y), derive the expression for Corr(X, Y) using Var(X + Y), Var(X), and Var(Y).
Assumptions
Assume that both random variables possess finite second moments, and all variances are computed under the same underlying probability model.