Question

Lines of regressions 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

• A. $\frac{b}{1-a}$
• B. $\frac{1-a}{b}$
• C. $\frac{\beta }{1-\beta }$
• D. $\frac{\alpha }{1-\alpha }$

Answer 1

The correct option is A $\frac{b}{1-a}$

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