Question

Let x &y be two variables With same mean. If the line of regression of y on x & x on y are respectively $y=ax+b$ & $x=\alpha y+\beta ,$ then value of common mean is

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

Answer 1

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

Given $\overline{x}=\overline{y}$ (i)
Since two lines of regressions pass through $(\overline{x},\overline{y})$ therefore $\overline{y}=a\overline{x}+b$
$\Rightarrow$ $\overline{x}=a\overline{x}+b$ (using (i))
$\Rightarrow$ $\overline{x}=\frac{b}{1-a}=\overline{y}$