Correlation coeffcient expressed in a different form

Question

If $X_j = a + bX_i$ and $X_i = a' + b'X_j$ how does one show that the correlation coefficient can be written as: $\rho_{i,j} = b \frac{\sigma_i}{\sigma_j}=b'\frac{\sigma_j}{\sigma_i}$ ?

Answer 1

Hint:

$b=\frac{cov(X,Y)}{\sigma^2_X}$ and $\rho_{ij}=\frac{cov(X,Y)}{\sigma_X \sigma_Y}$, then

$$ \rho_{ij}=\frac{cov(X,Y)}{\sigma_X \sigma_Y} = \frac{\sigma_X cov(X,Y)}{\sigma^2_X \sigma_Y} = b\frac{\sigma_X}{\sigma_Y} $$