# If you regress random variable Y against random variable X, would the results be the same if you regressed X against Y?

Nov 8, 2015

They would be the same only if the $V a r \left(x\right) = V a r \left(y\right)$

#### Explanation:

The Pearson correlation coefficient will always be the same regardless what the variances are. However, the regression lines will have different slopes (hence, not the same) unless the variances are equal.

This may be more technical than what you are looking for, but here is the slope of the regression line for both cases:

${\beta}_{1} = \frac{C o v \left(x , y\right)}{V a r \left(x\right)}$ This is regress y against x

${\beta}_{1} = \frac{C o v \left(x , y\right)}{V a r \left(y\right)}$ This is regress x against y

As you can see, they will only have the same slope if the variances are equal.

hope that helped