The error of prediction of x from the required line of regression is $n\sigma^2_x(1-\rho^2)$
This is given in my reference without any further explanation. What does this realy mean and how do we arrive such a conclusion ?
Things I know $$ \rho=\frac{Cov(X,Y)}{\sigma_x.\sigma_y}\\ $$
The regression line of $y$ on $x$ is $y-\bar{y}=\rho\dfrac{\sigma_y}{\sigma_x}(x-\bar{x})=b_{yx}(x-\bar{x})$ and The regression line of $x$ on $y$ is $x-\bar{x}=\rho\dfrac{\sigma_x}{\sigma_y}(y-\bar{y})=b_{xy}(y-\bar{y})$ $$ \rho^2=b_{yx}.b_{xy} $$ And I understand the regression line is derived by minimizing $s=\sum\epsilon^2=\sum\Big[(ax+b)-y\Big]^2$