Independent solution

How to solve this Covariance and Correlation question

Setup

Setup

Convert correlation to covariance using the two standard deviations.

σX=1,σY=2\sigma_X=1,\qquad \sigma_Y=2
Cov⁡(X,Y)=ρσXσY=12(1)(2)=1\operatorname{Cov}(X,Y)=\rho\sigma_X\sigma_Y=\frac12(1)(2)=1

Model

Model

Uncorrelated linear combinations have covariance zero. Expand bilinearly without using the means, which do not affect covariance.

Cov⁡(X+Y,cX+Y)=0\operatorname{Cov}(X+Y,cX+Y)=0

Compute

Compute

Collect the variance and cross-covariance terms that multiply c and the terms that do not.

cVar⁡(X)+Cov⁡(X,Y)+cCov⁡(Y,X)+Var⁡(Y)=0c\operatorname{Var}(X)+\operatorname{Cov}(X,Y)+c\operatorname{Cov}(Y,X)+\operatorname{Var}(Y)=0
c+1+c+4=0c+1+c+4=0
c=−52c=-\frac52

Answer

Answer

The required coefficient is negative five halves.

c=−52(B)\boxed{c=-\frac52\quad\text{(B)}}