# Correlation($U,V$)=Correlation($X,Y$) [closed]

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 $ac<0$.

• Have you tried anything? – angryavian Feb 16 '17 at 5:36
• Not yet. I'm really lost, so pointers to begin with would help a ton! – Amanda R. Feb 16 '17 at 5:51

Might be best to start with covariance and $a,c > 0$. $$Cov(U,V) = Cov(aX+b, cY+d) = Cov(aX,cY) + Cov(aX,d) + Cov(b,cY) + Cov(b,d)\\ = Cov(aX,cY) = acCov(X,Y).$$ Then $\rho(U,V) = Cor(U,V) = \frac{Cov(U,V)}{SD(U)SD(V)}.$
Finally, finish by finding $\sigma_U = SD(U)$ and $\sigma_V = SD(V).$
• $SD(U)=a\sigma_X$ and $SD(V)=c\sigma_Y$ right? – Amanda R. Feb 16 '17 at 7:42
• Right. $Var(aX) = a^2Var(X).$ Then take square root to get SD. – BruceET Feb 16 '17 at 7:56