Let the correlation coefficient between X and Y be $\rho$.
Show that the correlation coefficient between $aX + b$ and $cY + d$ can also be equal to $\rho$. Currently, I am trying to understand this concept to build a strong foundation in inferential statistics.
I understand that (and hopefully this is correct) $$\rho = Cov(X,Y)/\sigma_x\sigma_y$$ and that can eventually be algebraically deduced to be $$E[(X-E(X)/\sigma_x)]E[(Y-E(X)/\sigma_Y)] $$
Can someone explain to me how this can be related to the fact that $aX + b$ and $cY + d$ can also be equal to $\rho$?
Everything is linear making it quite easy.
$$\begin{split}\rho'&=\frac{Cov(aX+b, cY+d)}{\sigma_{aX+b}\sigma_{cY+d}}\\ &=\frac{acCov(X, Y)}{a\sigma_Xc\sigma_Y}\\ &=\frac{Cov(X,Y)}{\sigma_X\sigma_Y}\\ &=\rho\end{split}$$