Bound a Third Correlation from Two Known Correlations
Company: Tower
Role: Quantitative Researcher
Category: Statistics & Math
Difficulty: medium
Interview Round: Technical Screen
Three random variables have finite, positive variances. Their Pearson correlations satisfy corr(X1,X2) = 0.5 and corr(X2,X3) = 0.5. Determine the full possible range of corr(X1,X3).
### Clarifying Questions
May the joint distribution be singular? For the main answer, allow exact linear dependence, so attainable endpoints are included. No particular distribution family is required.
### What a Strong Answer Covers
Derive a necessary bound using a valid correlation matrix or residual variables, and show that every value in the proposed interval is attainable.
### Follow-up Questions
- How does the range change for general known correlations a and b?
- What changes if the correlation matrix must be positive definite?
Overview: Find the possible correlation between two variables that each correlate 0.5 with a third variable. Derive the bound from positive semidefiniteness and construct distributions that attain it.
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