Question

Prove that coefficient of correlation is the geometric mean of regression co-efficients.

Answer 1

We know that regression coeff. of $x$ on $y=b_{xy}=\rho \frac{{\sigma }_x}{{\sigma }_y}$ and regression coeff. of $y$ on $x=b_{yx}=\rho \frac{{\sigma }_y}{{\sigma }_x}$
$\therefore b_{xy}.b_{yx}=\rho \frac{{\sigma }_x}{{\sigma }_y}.\rho \frac{{\sigma }_y}{{\sigma }_x}={\rho }^2$
$\therefore \rho =\sqrt{b_{xy}.b_{yx}}$ (Here $\rho$ is correlation coefficient).