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How to compute correlation from variances
Problem Description
You are given two random variables, X and Y. You are provided with the following information:
- The variance of their sum: Var(X+Y)
- The variance of X: Var(X)
Questions
- Is this information sufficient to determine the correlation coefficient, ρ(X,Y)? If not, what additional quantity or quantities would be required?
- If you are also given Var(Y), derive the expression for ρ(X,Y) in terms of Var(X+Y), Var(X), and Var(Y).
- Consider a special case: if Var(Y) = Var(X), simplify the expression for ρ(X,Y) derived in the previous step.
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Core dilemma
The central challenge is to precisely apply foundational statistical definitions and properties (variance of a sum, covariance, correlation) to derive an unknown quantity from given information, demonstrating rigorous algebraic manipulation and conceptual understanding.
Mid-level signals3
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How to compute correlation from variances
Problem Description
You are given two random variables, X and Y. You are provided with the following information:
- The variance of their sum: Var(X+Y)
- The variance of X: Var(X)
Questions
- Is this information sufficient to determine the correlation coefficient, ρ(X,Y)? If not, what additional quantity or quantities would be required?
- If you are also given Var(Y), derive the expression for ρ(X,Y) in terms of Var(X+Y), Var(X), and Var(Y).
- Consider a special case: if Var(Y) = Var(X), simplify the expression for ρ(X,Y) derived in the previous step.