Proof that given 2 variables X and Y with correlation $\rho_{X,Y}$ and given $U=a+bX$ and $V=c+dY$ then $\rho_{X,Y} = \rho_{U,V} $ if $bd > 0$

Question

Given 2 random continuous variables X and Y with correlation $\rho_{X,Y}$ and given $U=a+bX$ and $V=c+dY$

Prove that

$ \rho_{X,Y}= \left\{ \begin{array}{lc} \rho_{U,V} & if &bd > 0 \\ \\ -\rho_{U,V} & if &bd < 0 \end{array} \right.$

I know that $\rho_{X,Y}$ = $\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}=\frac{E(X,Y)-E(X)E(Y)}{\sqrt{(E(X^2)-E^2(X))(E(Y^2)-E^2(Y))}}=\frac{E(X,Y)-E(X)E(Y)}{\sqrt{E(X^2)E(Y^2)-E(X^2)E^2(Y)-E^2(X)E(Y^2)+E^2(X)E^2(Y)}}$

But I do not know how to proceed, can I get some help? Thanks.

Answer 1

$$\text{Cov}(U, V) = \text{Cov}(a+bX, c+dY) = \text{[some expression involving $\text{Cov}(X,Y)$]}$$

$$\text{Var}(U) = \text{Var}(a+bX) = \text{[some expression involving $\text{Var}(X)$]}$$

$$\text{Var}(V) = \text{Var}(c+dY) = \text{[some expression involving $\text{Var}(Y)$]}$$

Use what you know about the definition of covariance and variance to fill in the blanks on the right-hand side. Then relate all this back to $\rho_{U,V}$ and $\rho_{X,Y}$.