Question
For bivariate data, the regression coefficient of Y on X is 0.4 and the regression coefficient of X on Y is 0.9. Find the value of the variance of Y if the variance of X is 9.

Answer

Given, $b_{y x}=0.4, b_{x y}=0.9, \sigma_x^2=9, \sigma_x=3$
$
\begin{aligned}
& r^2=b_{y x} \cdot b_{x y} \\
& r^2=0.4 \times 0.9 \\
& r^2=0.36 \\
& r= \pm 0.6
\end{aligned}
$
Since $b_{y x}$ and $b_{x y}$ are positive $\therefore r=0.6$
$
b_{y x}=\frac{r \cdot \sigma_y}{\sigma_x}
$
$
\begin{aligned}
& 0.4=0.6 \times \frac{\sigma_y}{3} \\
& \frac{4}{10}=\frac{6}{10} \times \frac{\sigma_y}{3} \\
& \sigma_y=2 \\
& \therefore \sigma_y{ }^2=4
\end{aligned}
$
$\therefore$ Variance of $y$ is 4

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