Correlation relation between a normal random vector $(X,Y)$ and its standarizatio $(X',Y')$

Question

Suppose we have a normal random vector $(X,Y)$ with correlation $\rho$.

If we define $X=\mu_X+\sigma_X X'$ and $Y=\mu_Y+\sigma_Y Y'$, where $\mu$ is the expected value and $\sigma$ is the standard deviation, what would be $\rho(X',Y')$?

Is it $$\rho(X,Y)=\rho(X',Y')?$$

Answer 1

Hint:

Verify that $\mathsf{Cov}(aX+b,cY+d)=ac\mathsf{Cov}(X,Y)$ and (as a corollary) that $\mathsf{Var}(aX+b)=a^2\mathsf{Var}(X)$

Then draw conclusions for $\rho(aX+b,cY+d)$.