Correlation($U,V$)=Correlation($X,Y$) Let $X$ and $Y$ be random variables such that $0<\sigma^2_X<\infty$ and $0<\sigma^2_Y<\infty$. Suppose that $U=aX+b$ and $V=cY+d$, where $a \not= 0$ and $c \not= 0$. Show that $\rho(U,V)=\rho(X,Y)$ if $ac>0$, and $\rho(U,V)=-\rho(X,Y)$ if $ac<0$.

\begin{eqnarray*} \displaystyle\rho(U, V) & = & Cov\left(\frac{U-\mathbb{E}(U)}{\sigma_U}, \frac{V-\mathbb{E}(V)}{\sigma_V}\right) \\ & = & Cov\left(\frac{aX+b-a\mathbb{E}(X)-b}{|a|\sigma_X}, \frac{cY+d-c\mathbb{E}(Y)-d}{|c|\sigma_Y}\right) \\ & = & \frac{ac}{|a||c|}Cov\left(\frac{X-\mathbb{E}(X)}{\sigma_X}, \frac{Y-\mathbb{E}(Y)}{\sigma_Y}\right) \\ & = & \frac{ac}{|a||c|}\rho(X, Y) \\ & = & \begin{cases} \rho(X, Y) & \text{ if } ac > 0 \\ -\rho(X, Y) & \text{ if } ac < 0 \end{cases} \end{eqnarray*}
