Correlation Under Affine Maps

Since $\operatorname{Corr}(X,Y)$ is given, $X$ and $Y$ have finite, positive variances. Write $\sigma_X^2=\operatorname{Var}(X)$ and $\sigma_Y^2=\operatorname{Var}(Y)$ .

Key idea. Additive constants do not change covariances or variances. Multiplicative constants come out of each, so only their signs survive in the ratio.

For constants $a,c\neq 0$ and $b,d$ , bilinearity of covariance gives

\operatorname{Cov}(aX+b,\,cY+d)=ac\operatorname{Cov}(X,Y).

The variances transform as

\operatorname{Var}(aX+b)=a^2\sigma_X^2, \qquad \operatorname{Var}(cY+d)=c^2\sigma_Y^2.

Dividing the covariance by the product of the standard deviations,

\operatorname{Corr}(aX+b,\,cY+d)=\frac{ac\operatorname{Cov}(X,Y)}{|a|\,|c|\,\sigma_X\sigma_Y} =\operatorname{sign}(ac)\operatorname{Corr}(X,Y).

Here $3X+1$ has $a=3$ , and $1-Y=-Y+1$ has $c=-1$ . Hence $\operatorname{sign}(ac)=-1$ and

\operatorname{Corr}(3X+1,\,1-Y)=-(-0.4)=0.4.

The quantity is therefore determined by the given information:

\boxed{\operatorname{Corr}(3X+1,\,1-Y)=\tfrac{2}{5}}