Independent solution

How to solve this Covariance and Correlation question

Answer in brief

The supplied correlation and standard deviations give covariance 1. Expanding the covariance of the two linear combinations yields 2c + 5, so c = -5/2 and choice B.

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)}}