How to compute correlation from variances
Company: Jefferies
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Technical Screen
You have two random variables \(X\) and \(Y\).
You are told:
- \(\mathrm{Var}(X+Y)\)
- \(\mathrm{Var}(X)\)
1) Is this information sufficient to determine the correlation \(\rho_{X,Y}\)? If not, what additional quantity(ies) 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 \(\rho_{X,Y}\).
Quick Answer: This question evaluates understanding of variance, covariance and their relationship to correlation, assessing facility with moment-based statistical algebra and linear dependence in the Statistics & Math domain for a Data Scientist role at a practical application level.