Browse Questions

# Lines of regression of $y$ on $x$ and $x$ on $y$ are respectively $y=ax+b$ and $x=\alpha y+ \beta$. If mean of $x$ and $y$ series is same then its value is

$\begin {array} {1 1} (A)\;\large\frac{b}{1-a} \: or \: \large\frac{\beta}{1- \alpha} & \quad (B)\;\large\frac{1-a}{b} \\ (C)\;\large\frac{\beta}{1-\beta} & \quad (D)\;\large\frac{\alpha}{1-\alpha} \end {array}$

Let mean of $x$ and $y$ series are $\overline x$ and $\overline y$
Lines of regression pass through mean $( \overline x , \overline y )$ hence
$\overline y = a \overline x +b$
$\overline x = \alpha \overline y + \beta$
Given that $\overline x = \overline y$
$\Rightarrow \large\frac{b}{1-a} = \large\frac{ \beta}{1- \alpha}$
Ans : (A)