Removing the X component to decorrelate Y

Question

Random variables $X$ and $Y$ satisfy $\operatorname{Var}(X)=16$ and $\operatorname{Cov}(X,Y)=8$ . Which constant $k$ makes $X$ uncorrelated with $Y-kX$ ?

Short answer

$k=\tfrac12$ . Covariance is bilinear, so $\operatorname{Cov}(X,Y-kX)=\operatorname{Cov}(X,Y)-k\operatorname{Var}(X)=8-16k$ , which is zero when $k=\tfrac12$ . In general $k$ is the regression slope $\operatorname{Cov}(X,Y)/\operatorname{Var}(X)$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Covariance and Correlation
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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